Properties

Label 65.2.b.a
Level $65$
Weight $2$
Character orbit 65.b
Analytic conductor $0.519$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,2,Mod(14,65)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.b (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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 1.45161i
−0.854638 0.854638i
0.403032 + 0.403032i
2.67513i 0.481194i −5.15633 1.67513 + 1.48119i −1.28726 0.806063i 8.44358i 2.76845 3.96239 4.48119i
14.2 1.53919i 3.17009i −0.369102 0.539189 2.17009i 4.87936 1.70928i 2.51026i −7.04945 −3.34017 0.829914i
14.3 1.21432i 1.31111i 0.525428 −2.21432 + 0.311108i −1.59210 2.90321i 3.06668i 1.28100 0.377784 + 2.68889i
14.4 1.21432i 1.31111i 0.525428 −2.21432 0.311108i −1.59210 2.90321i 3.06668i 1.28100 0.377784 2.68889i
14.5 1.53919i 3.17009i −0.369102 0.539189 + 2.17009i 4.87936 1.70928i 2.51026i −7.04945 −3.34017 + 0.829914i
14.6 2.67513i 0.481194i −5.15633 1.67513 1.48119i −1.28726 0.806063i 8.44358i 2.76845 3.96239 + 4.48119i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.b.a 6
3.b odd 2 1 585.2.c.b 6
4.b odd 2 1 1040.2.d.c 6
5.b even 2 1 inner 65.2.b.a 6
5.c odd 4 1 325.2.a.j 3
5.c odd 4 1 325.2.a.k 3
13.b even 2 1 845.2.b.c 6
13.c even 3 2 845.2.n.f 12
13.d odd 4 1 845.2.d.a 6
13.d odd 4 1 845.2.d.b 6
13.e even 6 2 845.2.n.g 12
13.f odd 12 2 845.2.l.d 12
13.f odd 12 2 845.2.l.e 12
15.d odd 2 1 585.2.c.b 6
15.e even 4 1 2925.2.a.bf 3
15.e even 4 1 2925.2.a.bj 3
20.d odd 2 1 1040.2.d.c 6
20.e even 4 1 5200.2.a.cb 3
20.e even 4 1 5200.2.a.cj 3
65.d even 2 1 845.2.b.c 6
65.g odd 4 1 845.2.d.a 6
65.g odd 4 1 845.2.d.b 6
65.h odd 4 1 4225.2.a.ba 3
65.h odd 4 1 4225.2.a.bh 3
65.l even 6 2 845.2.n.g 12
65.n even 6 2 845.2.n.f 12
65.s odd 12 2 845.2.l.d 12
65.s odd 12 2 845.2.l.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 1.a even 1 1 trivial
65.2.b.a 6 5.b even 2 1 inner
325.2.a.j 3 5.c odd 4 1
325.2.a.k 3 5.c odd 4 1
585.2.c.b 6 3.b odd 2 1
585.2.c.b 6 15.d odd 2 1
845.2.b.c 6 13.b even 2 1
845.2.b.c 6 65.d even 2 1
845.2.d.a 6 13.d odd 4 1
845.2.d.a 6 65.g odd 4 1
845.2.d.b 6 13.d odd 4 1
845.2.d.b 6 65.g odd 4 1
845.2.l.d 12 13.f odd 12 2
845.2.l.d 12 65.s odd 12 2
845.2.l.e 12 13.f odd 12 2
845.2.l.e 12 65.s odd 12 2
845.2.n.f 12 13.c even 3 2
845.2.n.f 12 65.n even 6 2
845.2.n.g 12 13.e even 6 2
845.2.n.g 12 65.l even 6 2
1040.2.d.c 6 4.b odd 2 1
1040.2.d.c 6 20.d odd 2 1
2925.2.a.bf 3 15.e even 4 1
2925.2.a.bj 3 15.e even 4 1
4225.2.a.ba 3 65.h odd 4 1
4225.2.a.bh 3 65.h odd 4 1
5200.2.a.cb 3 20.e even 4 1
5200.2.a.cj 3 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 31 T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + 20 T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} + 6 T^{2} + 8 T - 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 72 T^{4} + 1436 T^{2} + \cdots + 7396 \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 36 T - 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 10 T^{2} + 20 T - 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 32 T + 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + 5452 T^{2} + \cdots + 77284 \) Copy content Toggle raw display
$47$ \( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 40 T + 262)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 88 T - 754)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 248 T^{4} + 15568 T^{2} + \cdots + 55696 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + 24 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 180 T^{4} + 9200 T^{2} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} - 52 T - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 364 T^{4} + 12656 T^{2} + \cdots + 40000 \) Copy content Toggle raw display
show more
show less