Properties

Label 26.2.b.a
Level $26$
Weight $2$
Character orbit 26.b
Analytic conductor $0.208$
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] = [26,2,Mod(25,26)]
 
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([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
25.2 1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.2.b.a 2
3.b odd 2 1 234.2.b.b 2
4.b odd 2 1 208.2.f.a 2
5.b even 2 1 650.2.d.b 2
5.c odd 4 1 650.2.c.a 2
5.c odd 4 1 650.2.c.d 2
7.b odd 2 1 1274.2.d.c 2
7.c even 3 2 1274.2.n.d 4
7.d odd 6 2 1274.2.n.c 4
8.b even 2 1 832.2.f.d 2
8.d odd 2 1 832.2.f.b 2
12.b even 2 1 1872.2.c.f 2
13.b even 2 1 inner 26.2.b.a 2
13.c even 3 2 338.2.e.c 4
13.d odd 4 1 338.2.a.b 1
13.d odd 4 1 338.2.a.d 1
13.e even 6 2 338.2.e.c 4
13.f odd 12 2 338.2.c.b 2
13.f odd 12 2 338.2.c.f 2
39.d odd 2 1 234.2.b.b 2
39.f even 4 1 3042.2.a.g 1
39.f even 4 1 3042.2.a.j 1
52.b odd 2 1 208.2.f.a 2
52.f even 4 1 2704.2.a.j 1
52.f even 4 1 2704.2.a.k 1
65.d even 2 1 650.2.d.b 2
65.g odd 4 1 8450.2.a.h 1
65.g odd 4 1 8450.2.a.u 1
65.h odd 4 1 650.2.c.a 2
65.h odd 4 1 650.2.c.d 2
91.b odd 2 1 1274.2.d.c 2
91.r even 6 2 1274.2.n.d 4
91.s odd 6 2 1274.2.n.c 4
104.e even 2 1 832.2.f.d 2
104.h odd 2 1 832.2.f.b 2
156.h even 2 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 1.a even 1 1 trivial
26.2.b.a 2 13.b even 2 1 inner
208.2.f.a 2 4.b odd 2 1
208.2.f.a 2 52.b odd 2 1
234.2.b.b 2 3.b odd 2 1
234.2.b.b 2 39.d odd 2 1
338.2.a.b 1 13.d odd 4 1
338.2.a.d 1 13.d odd 4 1
338.2.c.b 2 13.f odd 12 2
338.2.c.f 2 13.f odd 12 2
338.2.e.c 4 13.c even 3 2
338.2.e.c 4 13.e even 6 2
650.2.c.a 2 5.c odd 4 1
650.2.c.a 2 65.h odd 4 1
650.2.c.d 2 5.c odd 4 1
650.2.c.d 2 65.h odd 4 1
650.2.d.b 2 5.b even 2 1
650.2.d.b 2 65.d even 2 1
832.2.f.b 2 8.d odd 2 1
832.2.f.b 2 104.h odd 2 1
832.2.f.d 2 8.b even 2 1
832.2.f.d 2 104.e even 2 1
1274.2.d.c 2 7.b odd 2 1
1274.2.d.c 2 91.b odd 2 1
1274.2.n.c 4 7.d odd 6 2
1274.2.n.c 4 91.s odd 6 2
1274.2.n.d 4 7.c even 3 2
1274.2.n.d 4 91.r even 6 2
1872.2.c.f 2 12.b even 2 1
1872.2.c.f 2 156.h even 2 1
2704.2.a.j 1 52.f even 4 1
2704.2.a.k 1 52.f even 4 1
3042.2.a.g 1 39.f even 4 1
3042.2.a.j 1 39.f even 4 1
8450.2.a.h 1 65.g odd 4 1
8450.2.a.u 1 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(26, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 13 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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