Properties

Label 26.4.a.c
Level $26$
Weight $4$
Character orbit 26.a
Self dual yes
Analytic conductor $1.534$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,4,Mod(1,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, 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("26.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(1.53404966015\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 18 q^{5} + 8 q^{6} + 20 q^{7} + 8 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 18 q^{5} + 8 q^{6} + 20 q^{7} + 8 q^{8} - 11 q^{9} - 36 q^{10} - 48 q^{11} + 16 q^{12} + 13 q^{13} + 40 q^{14} - 72 q^{15} + 16 q^{16} + 66 q^{17} - 22 q^{18} - 16 q^{19} - 72 q^{20} + 80 q^{21} - 96 q^{22} + 168 q^{23} + 32 q^{24} + 199 q^{25} + 26 q^{26} - 152 q^{27} + 80 q^{28} + 6 q^{29} - 144 q^{30} + 20 q^{31} + 32 q^{32} - 192 q^{33} + 132 q^{34} - 360 q^{35} - 44 q^{36} + 254 q^{37} - 32 q^{38} + 52 q^{39} - 144 q^{40} - 390 q^{41} + 160 q^{42} - 124 q^{43} - 192 q^{44} + 198 q^{45} + 336 q^{46} - 468 q^{47} + 64 q^{48} + 57 q^{49} + 398 q^{50} + 264 q^{51} + 52 q^{52} + 558 q^{53} - 304 q^{54} + 864 q^{55} + 160 q^{56} - 64 q^{57} + 12 q^{58} - 96 q^{59} - 288 q^{60} - 826 q^{61} + 40 q^{62} - 220 q^{63} + 64 q^{64} - 234 q^{65} - 384 q^{66} - 160 q^{67} + 264 q^{68} + 672 q^{69} - 720 q^{70} - 420 q^{71} - 88 q^{72} + 362 q^{73} + 508 q^{74} + 796 q^{75} - 64 q^{76} - 960 q^{77} + 104 q^{78} + 776 q^{79} - 288 q^{80} - 311 q^{81} - 780 q^{82} + 320 q^{84} - 1188 q^{85} - 248 q^{86} + 24 q^{87} - 384 q^{88} + 1626 q^{89} + 396 q^{90} + 260 q^{91} + 672 q^{92} + 80 q^{93} - 936 q^{94} + 288 q^{95} + 128 q^{96} - 1294 q^{97} + 114 q^{98} + 528 q^{99}+O(q^{100}) \) 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
2.00000 4.00000 4.00000 −18.0000 8.00000 20.0000 8.00000 −11.0000 −36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( -1 \)

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 26.4.a.c 1
3.b odd 2 1 234.4.a.e 1
4.b odd 2 1 208.4.a.b 1
5.b even 2 1 650.4.a.b 1
5.c odd 4 2 650.4.b.f 2
7.b odd 2 1 1274.4.a.d 1
8.b even 2 1 832.4.a.d 1
8.d odd 2 1 832.4.a.o 1
12.b even 2 1 1872.4.a.q 1
13.b even 2 1 338.4.a.c 1
13.c even 3 2 338.4.c.a 2
13.d odd 4 2 338.4.b.d 2
13.e even 6 2 338.4.c.e 2
13.f odd 12 4 338.4.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 1.a even 1 1 trivial
208.4.a.b 1 4.b odd 2 1
234.4.a.e 1 3.b odd 2 1
338.4.a.c 1 13.b even 2 1
338.4.b.d 2 13.d odd 4 2
338.4.c.a 2 13.c even 3 2
338.4.c.e 2 13.e even 6 2
338.4.e.a 4 13.f odd 12 4
650.4.a.b 1 5.b even 2 1
650.4.b.f 2 5.c odd 4 2
832.4.a.d 1 8.b even 2 1
832.4.a.o 1 8.d odd 2 1
1274.4.a.d 1 7.b odd 2 1
1872.4.a.q 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(26))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T + 18 \) Copy content Toggle raw display
$7$ \( T - 20 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 66 \) Copy content Toggle raw display
$19$ \( T + 16 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 20 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 390 \) Copy content Toggle raw display
$43$ \( T + 124 \) Copy content Toggle raw display
$47$ \( T + 468 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T + 96 \) Copy content Toggle raw display
$61$ \( T + 826 \) Copy content Toggle raw display
$67$ \( T + 160 \) Copy content Toggle raw display
$71$ \( T + 420 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 776 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 1626 \) Copy content Toggle raw display
$97$ \( T + 1294 \) Copy content Toggle raw display
show more
show less