Properties

Label 13.4.a.a
Level $13$
Weight $4$
Character orbit 13.a
Self dual yes
Analytic conductor $0.767$
Analytic rank $1$
Dimension $1$
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] = [13,4,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(0.767024830075\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} - 7 q^{3} + 17 q^{4} - 7 q^{5} + 35 q^{6} - 13 q^{7} - 45 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - 7 q^{3} + 17 q^{4} - 7 q^{5} + 35 q^{6} - 13 q^{7} - 45 q^{8} + 22 q^{9} + 35 q^{10} - 26 q^{11} - 119 q^{12} + 13 q^{13} + 65 q^{14} + 49 q^{15} + 89 q^{16} + 77 q^{17} - 110 q^{18} - 126 q^{19} - 119 q^{20} + 91 q^{21} + 130 q^{22} - 96 q^{23} + 315 q^{24} - 76 q^{25} - 65 q^{26} + 35 q^{27} - 221 q^{28} - 82 q^{29} - 245 q^{30} + 196 q^{31} - 85 q^{32} + 182 q^{33} - 385 q^{34} + 91 q^{35} + 374 q^{36} - 131 q^{37} + 630 q^{38} - 91 q^{39} + 315 q^{40} + 336 q^{41} - 455 q^{42} - 201 q^{43} - 442 q^{44} - 154 q^{45} + 480 q^{46} - 105 q^{47} - 623 q^{48} - 174 q^{49} + 380 q^{50} - 539 q^{51} + 221 q^{52} - 432 q^{53} - 175 q^{54} + 182 q^{55} + 585 q^{56} + 882 q^{57} + 410 q^{58} - 294 q^{59} + 833 q^{60} - 56 q^{61} - 980 q^{62} - 286 q^{63} - 287 q^{64} - 91 q^{65} - 910 q^{66} + 478 q^{67} + 1309 q^{68} + 672 q^{69} - 455 q^{70} + 9 q^{71} - 990 q^{72} + 98 q^{73} + 655 q^{74} + 532 q^{75} - 2142 q^{76} + 338 q^{77} + 455 q^{78} + 1304 q^{79} - 623 q^{80} - 839 q^{81} - 1680 q^{82} - 308 q^{83} + 1547 q^{84} - 539 q^{85} + 1005 q^{86} + 574 q^{87} + 1170 q^{88} - 1190 q^{89} + 770 q^{90} - 169 q^{91} - 1632 q^{92} - 1372 q^{93} + 525 q^{94} + 882 q^{95} + 595 q^{96} + 70 q^{97} + 870 q^{98} - 572 q^{99}+O(q^{100}) \) 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
0
−5.00000 −7.00000 17.0000 −7.00000 35.0000 −13.0000 −45.0000 22.0000 35.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

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 13.4.a.a 1
3.b odd 2 1 117.4.a.b 1
4.b odd 2 1 208.4.a.g 1
5.b even 2 1 325.4.a.d 1
5.c odd 4 2 325.4.b.b 2
7.b odd 2 1 637.4.a.a 1
8.b even 2 1 832.4.a.r 1
8.d odd 2 1 832.4.a.a 1
11.b odd 2 1 1573.4.a.a 1
12.b even 2 1 1872.4.a.k 1
13.b even 2 1 169.4.a.e 1
13.c even 3 2 169.4.c.e 2
13.d odd 4 2 169.4.b.a 2
13.e even 6 2 169.4.c.a 2
13.f odd 12 4 169.4.e.e 4
39.d odd 2 1 1521.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 1.a even 1 1 trivial
117.4.a.b 1 3.b odd 2 1
169.4.a.e 1 13.b even 2 1
169.4.b.a 2 13.d odd 4 2
169.4.c.a 2 13.e even 6 2
169.4.c.e 2 13.c even 3 2
169.4.e.e 4 13.f odd 12 4
208.4.a.g 1 4.b odd 2 1
325.4.a.d 1 5.b even 2 1
325.4.b.b 2 5.c odd 4 2
637.4.a.a 1 7.b odd 2 1
832.4.a.a 1 8.d odd 2 1
832.4.a.r 1 8.b even 2 1
1521.4.a.a 1 39.d odd 2 1
1573.4.a.a 1 11.b odd 2 1
1872.4.a.k 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T + 7 \) Copy content Toggle raw display
$7$ \( T + 13 \) Copy content Toggle raw display
$11$ \( T + 26 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 77 \) Copy content Toggle raw display
$19$ \( T + 126 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T + 82 \) Copy content Toggle raw display
$31$ \( T - 196 \) Copy content Toggle raw display
$37$ \( T + 131 \) Copy content Toggle raw display
$41$ \( T - 336 \) Copy content Toggle raw display
$43$ \( T + 201 \) Copy content Toggle raw display
$47$ \( T + 105 \) Copy content Toggle raw display
$53$ \( T + 432 \) Copy content Toggle raw display
$59$ \( T + 294 \) Copy content Toggle raw display
$61$ \( T + 56 \) Copy content Toggle raw display
$67$ \( T - 478 \) Copy content Toggle raw display
$71$ \( T - 9 \) Copy content Toggle raw display
$73$ \( T - 98 \) Copy content Toggle raw display
$79$ \( T - 1304 \) Copy content Toggle raw display
$83$ \( T + 308 \) Copy content Toggle raw display
$89$ \( T + 1190 \) Copy content Toggle raw display
$97$ \( T - 70 \) Copy content Toggle raw display
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