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Properties

Label 13.2.e.a
Level $13$
Weight $2$
Character orbit 13.e
Analytic conductor $0.104$
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] = [13,2,Mod(4,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), 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: \(0.103805522628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−1.50000 − 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i − 1.73205i 3.00000 − 1.73205i 0 1.73205i −0.500000 − 0.866025i −1.50000 + 2.59808i
10.1 −1.50000 + 0.866025i −1.00000 − 1.73205i 0.500000 − 0.866025i 1.73205i 3.00000 + 1.73205i 0 − 1.73205i −0.500000 + 0.866025i −1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.2.e.a ✓ 2
3.b odd 2 1 117.2.q.c 2
4.b odd 2 1 208.2.w.b 2
5.b even 2 1 325.2.n.a 2
5.c odd 4 2 325.2.m.a 4
7.b odd 2 1 637.2.q.a 2
7.c even 3 1 637.2.k.a 2
7.c even 3 1 637.2.u.c 2
7.d odd 6 1 637.2.k.c 2
7.d odd 6 1 637.2.u.b 2
8.b even 2 1 832.2.w.d 2
8.d odd 2 1 832.2.w.a 2
12.b even 2 1 1872.2.by.d 2
13.b even 2 1 169.2.e.a 2
13.c even 3 1 169.2.b.a 2
13.c even 3 1 169.2.e.a 2
13.d odd 4 2 169.2.c.a 4
13.e even 6 1 inner 13.2.e.a ✓ 2
13.e even 6 1 169.2.b.a 2
13.f odd 12 2 169.2.a.a 2
13.f odd 12 2 169.2.c.a 4
39.h odd 6 1 117.2.q.c 2
39.h odd 6 1 1521.2.b.a 2
39.i odd 6 1 1521.2.b.a 2
39.k even 12 2 1521.2.a.k 2
52.i odd 6 1 208.2.w.b 2
52.i odd 6 1 2704.2.f.b 2
52.j odd 6 1 2704.2.f.b 2
52.l even 12 2 2704.2.a.o 2
65.l even 6 1 325.2.n.a 2
65.r odd 12 2 325.2.m.a 4
65.s odd 12 2 4225.2.a.v 2
91.k even 6 1 637.2.u.c 2
91.l odd 6 1 637.2.u.b 2
91.p odd 6 1 637.2.k.c 2
91.t odd 6 1 637.2.q.a 2
91.u even 6 1 637.2.k.a 2
91.bc even 12 2 8281.2.a.q 2
104.p odd 6 1 832.2.w.a 2
104.s even 6 1 832.2.w.d 2
156.r even 6 1 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a ✓ 2 1.a even 1 1 trivial
13.2.e.a ✓ 2 13.e even 6 1 inner
117.2.q.c 2 3.b odd 2 1
117.2.q.c 2 39.h odd 6 1
169.2.a.a 2 13.f odd 12 2
169.2.b.a 2 13.c even 3 1
169.2.b.a 2 13.e even 6 1
169.2.c.a 4 13.d odd 4 2
169.2.c.a 4 13.f odd 12 2
169.2.e.a 2 13.b even 2 1
169.2.e.a 2 13.c even 3 1
208.2.w.b 2 4.b odd 2 1
208.2.w.b 2 52.i odd 6 1
325.2.m.a 4 5.c odd 4 2
325.2.m.a 4 65.r odd 12 2
325.2.n.a 2 5.b even 2 1
325.2.n.a 2 65.l even 6 1
637.2.k.a 2 7.c even 3 1
637.2.k.a 2 91.u even 6 1
637.2.k.c 2 7.d odd 6 1
637.2.k.c 2 91.p odd 6 1
637.2.q.a 2 7.b odd 2 1
637.2.q.a 2 91.t odd 6 1
637.2.u.b 2 7.d odd 6 1
637.2.u.b 2 91.l odd 6 1
637.2.u.c 2 7.c even 3 1
637.2.u.c 2 91.k even 6 1
832.2.w.a 2 8.d odd 2 1
832.2.w.a 2 104.p odd 6 1
832.2.w.d 2 8.b even 2 1
832.2.w.d 2 104.s even 6 1
1521.2.a.k 2 39.k even 12 2
1521.2.b.a 2 39.h odd 6 1
1521.2.b.a 2 39.i odd 6 1
1872.2.by.d 2 12.b even 2 1
1872.2.by.d 2 156.r even 6 1
2704.2.a.o 2 52.l even 12 2
2704.2.f.b 2 52.i odd 6 1
2704.2.f.b 2 52.j odd 6 1
4225.2.a.v 2 65.s odd 12 2
8281.2.a.q 2 91.bc even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
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