Properties

Label 67.2.a.b
Level $67$
Weight $2$
Character orbit 67.a
Self dual yes
Analytic conductor $0.535$
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] = [67,2,Mod(1,67)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(67, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("67.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 67.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: \(0.534997693543\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (\beta - 2) q^{3} + 3 \beta q^{4} - 3 q^{5} + q^{6} + ( - 3 \beta + 1) q^{7} + ( - 4 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (\beta - 2) q^{3} + 3 \beta q^{4} - 3 q^{5} + q^{6} + ( - 3 \beta + 1) q^{7} + ( - 4 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9} + (3 \beta + 3) q^{10} + (2 \beta - 1) q^{11} + ( - 3 \beta + 3) q^{12} + (3 \beta - 5) q^{13} + (5 \beta + 2) q^{14} + ( - 3 \beta + 6) q^{15} + (3 \beta + 5) q^{16} + (2 \beta - 4) q^{17} + (4 \beta + 1) q^{18} + ( - 3 \beta + 2) q^{19} - 9 \beta q^{20} + (4 \beta - 5) q^{21} + ( - 3 \beta - 1) q^{22} + (4 \beta + 1) q^{23} + (3 \beta - 2) q^{24} + 4 q^{25} + ( - \beta + 2) q^{26} + (2 \beta - 1) q^{27} + ( - 6 \beta - 9) q^{28} + ( - 4 \beta - 1) q^{29} - 3 q^{30} - q^{31} + ( - 3 \beta - 6) q^{32} + ( - 3 \beta + 4) q^{33} + 2 q^{34} + (9 \beta - 3) q^{35} + ( - 3 \beta - 9) q^{36} + ( - 3 \beta + 1) q^{37} + (4 \beta + 1) q^{38} + ( - 8 \beta + 13) q^{39} + (12 \beta + 3) q^{40} + (\beta - 2) q^{41} + ( - 3 \beta + 1) q^{42} + 3 \beta q^{43} + (3 \beta + 6) q^{44} + (9 \beta - 6) q^{45} + ( - 9 \beta - 5) q^{46} + ( - \beta - 7) q^{47} + (2 \beta - 7) q^{48} + (3 \beta + 3) q^{49} + ( - 4 \beta - 4) q^{50} + ( - 6 \beta + 10) q^{51} + ( - 6 \beta + 9) q^{52} - 9 q^{53} + ( - 3 \beta - 1) q^{54} + ( - 6 \beta + 3) q^{55} + (11 \beta + 11) q^{56} + (5 \beta - 7) q^{57} + (9 \beta + 5) q^{58} + 6 q^{59} + (9 \beta - 9) q^{60} + ( - 9 \beta + 1) q^{61} + (\beta + 1) q^{62} + 11 q^{63} + (6 \beta - 1) q^{64} + ( - 9 \beta + 15) q^{65} + (2 \beta - 1) q^{66} - q^{67} + ( - 6 \beta + 6) q^{68} + ( - 3 \beta + 2) q^{69} + ( - 15 \beta - 6) q^{70} + ( - 2 \beta + 7) q^{71} + (7 \beta + 10) q^{72} - 4 q^{73} + (5 \beta + 2) q^{74} + (4 \beta - 8) q^{75} + ( - 3 \beta - 9) q^{76} + ( - \beta - 7) q^{77} + (3 \beta - 5) q^{78} + (9 \beta - 8) q^{79} + ( - 9 \beta - 15) q^{80} + (6 \beta - 2) q^{81} + q^{82} + ( - 7 \beta - 4) q^{83} + ( - 3 \beta + 12) q^{84} + ( - 6 \beta + 12) q^{85} + ( - 6 \beta - 3) q^{86} + (3 \beta - 2) q^{87} + ( - 6 \beta - 7) q^{88} + ( - 2 \beta + 1) q^{89} + ( - 12 \beta - 3) q^{90} + (9 \beta - 14) q^{91} + (15 \beta + 12) q^{92} + ( - \beta + 2) q^{93} + (9 \beta + 8) q^{94} + (9 \beta - 6) q^{95} + ( - 3 \beta + 9) q^{96} + (12 \beta - 5) q^{97} + ( - 9 \beta - 6) q^{98} + (\beta - 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9} + 9 q^{10} + 3 q^{12} - 7 q^{13} + 9 q^{14} + 9 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} + q^{19} - 9 q^{20} - 6 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 3 q^{26} - 24 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} - 15 q^{32} + 5 q^{33} + 4 q^{34} + 3 q^{35} - 21 q^{36} - q^{37} + 6 q^{38} + 18 q^{39} + 18 q^{40} - 3 q^{41} - q^{42} + 3 q^{43} + 15 q^{44} - 3 q^{45} - 19 q^{46} - 15 q^{47} - 12 q^{48} + 9 q^{49} - 12 q^{50} + 14 q^{51} + 12 q^{52} - 18 q^{53} - 5 q^{54} + 33 q^{56} - 9 q^{57} + 19 q^{58} + 12 q^{59} - 9 q^{60} - 7 q^{61} + 3 q^{62} + 22 q^{63} + 4 q^{64} + 21 q^{65} - 2 q^{67} + 6 q^{68} + q^{69} - 27 q^{70} + 12 q^{71} + 27 q^{72} - 8 q^{73} + 9 q^{74} - 12 q^{75} - 21 q^{76} - 15 q^{77} - 7 q^{78} - 7 q^{79} - 39 q^{80} + 2 q^{81} + 2 q^{82} - 15 q^{83} + 21 q^{84} + 18 q^{85} - 12 q^{86} - q^{87} - 20 q^{88} - 18 q^{90} - 19 q^{91} + 39 q^{92} + 3 q^{93} + 25 q^{94} - 3 q^{95} + 15 q^{96} + 2 q^{97} - 21 q^{98} - 15 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 −0.381966 4.85410 −3.00000 1.00000 −3.85410 −7.47214 −2.85410 7.85410
1.2 −0.381966 −2.61803 −1.85410 −3.00000 1.00000 2.85410 1.47214 3.85410 1.14590
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(67\) \(1\)

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 67.2.a.b 2
3.b odd 2 1 603.2.a.h 2
4.b odd 2 1 1072.2.a.f 2
5.b even 2 1 1675.2.a.h 2
5.c odd 4 2 1675.2.c.d 4
7.b odd 2 1 3283.2.a.f 2
8.b even 2 1 4288.2.a.p 2
8.d odd 2 1 4288.2.a.h 2
11.b odd 2 1 8107.2.a.i 2
12.b even 2 1 9648.2.a.bi 2
67.b odd 2 1 4489.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.a.b 2 1.a even 1 1 trivial
603.2.a.h 2 3.b odd 2 1
1072.2.a.f 2 4.b odd 2 1
1675.2.a.h 2 5.b even 2 1
1675.2.c.d 4 5.c odd 4 2
3283.2.a.f 2 7.b odd 2 1
4288.2.a.h 2 8.d odd 2 1
4288.2.a.p 2 8.b even 2 1
4489.2.a.d 2 67.b odd 2 1
8107.2.a.i 2 11.b odd 2 1
9648.2.a.bi 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$11$ \( T^{2} - 5 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 89 \) Copy content Toggle raw display
$67$ \( (T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 31 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 7T - 89 \) Copy content Toggle raw display
$83$ \( T^{2} + 15T - 5 \) Copy content Toggle raw display
$89$ \( T^{2} - 5 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 179 \) Copy content Toggle raw display
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