Properties

Label 261.2.c.a
Level $261$
Weight $2$
Character orbit 261.c
Analytic conductor $2.084$
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] = [261,2,Mod(28,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.c (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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 3 q^{4} + 3 q^{5} + 2 q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 3 q^{4} + 3 q^{5} + 2 q^{7} - \beta q^{8} + 3 \beta q^{10} + \beta q^{11} - q^{13} + 2 \beta q^{14} - q^{16} - 2 \beta q^{17} - 9 q^{20} - 5 q^{22} - 6 q^{23} + 4 q^{25} - \beta q^{26} - 6 q^{28} + (2 \beta + 3) q^{29} - 3 \beta q^{31} - 3 \beta q^{32} + 10 q^{34} + 6 q^{35} - 3 \beta q^{40} - 2 \beta q^{41} + 3 \beta q^{43} - 3 \beta q^{44} - 6 \beta q^{46} + \beta q^{47} - 3 q^{49} + 4 \beta q^{50} + 3 q^{52} + 9 q^{53} + 3 \beta q^{55} - 2 \beta q^{56} + (3 \beta - 10) q^{58} - 6 q^{59} - 6 \beta q^{61} + 15 q^{62} + 13 q^{64} - 3 q^{65} + 8 q^{67} + 6 \beta q^{68} + 6 \beta q^{70} + 2 \beta q^{77} + 3 \beta q^{79} - 3 q^{80} + 10 q^{82} + 6 q^{83} - 6 \beta q^{85} - 15 q^{86} + 5 q^{88} - 2 \beta q^{89} - 2 q^{91} + 18 q^{92} - 5 q^{94} - 6 \beta q^{97} - 3 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{4} + 6 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{4} + 6 q^{5} + 4 q^{7} - 2 q^{13} - 2 q^{16} - 18 q^{20} - 10 q^{22} - 12 q^{23} + 8 q^{25} - 12 q^{28} + 6 q^{29} + 20 q^{34} + 12 q^{35} - 6 q^{49} + 6 q^{52} + 18 q^{53} - 20 q^{58} - 12 q^{59} + 30 q^{62} + 26 q^{64} - 6 q^{65} + 16 q^{67} - 6 q^{80} + 20 q^{82} + 12 q^{83} - 30 q^{86} + 10 q^{88} - 4 q^{91} + 36 q^{92} - 10 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
2.23607i
2.23607i
2.23607i 0 −3.00000 3.00000 0 2.00000 2.23607i 0 6.70820i
28.2 2.23607i 0 −3.00000 3.00000 0 2.00000 2.23607i 0 6.70820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.c.a 2
3.b odd 2 1 29.2.b.a 2
4.b odd 2 1 4176.2.o.k 2
12.b even 2 1 464.2.e.a 2
15.d odd 2 1 725.2.c.c 2
15.e even 4 2 725.2.d.a 4
21.c even 2 1 1421.2.b.b 2
24.f even 2 1 1856.2.e.f 2
24.h odd 2 1 1856.2.e.g 2
29.b even 2 1 inner 261.2.c.a 2
29.c odd 4 2 7569.2.a.i 2
87.d odd 2 1 29.2.b.a 2
87.f even 4 2 841.2.a.b 2
87.h odd 14 6 841.2.e.g 12
87.j odd 14 6 841.2.e.g 12
87.k even 28 12 841.2.d.h 12
116.d odd 2 1 4176.2.o.k 2
348.b even 2 1 464.2.e.a 2
435.b odd 2 1 725.2.c.c 2
435.p even 4 2 725.2.d.a 4
609.h even 2 1 1421.2.b.b 2
696.l even 2 1 1856.2.e.f 2
696.n odd 2 1 1856.2.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 3.b odd 2 1
29.2.b.a 2 87.d odd 2 1
261.2.c.a 2 1.a even 1 1 trivial
261.2.c.a 2 29.b even 2 1 inner
464.2.e.a 2 12.b even 2 1
464.2.e.a 2 348.b even 2 1
725.2.c.c 2 15.d odd 2 1
725.2.c.c 2 435.b odd 2 1
725.2.d.a 4 15.e even 4 2
725.2.d.a 4 435.p even 4 2
841.2.a.b 2 87.f even 4 2
841.2.d.h 12 87.k even 28 12
841.2.e.g 12 87.h odd 14 6
841.2.e.g 12 87.j odd 14 6
1421.2.b.b 2 21.c even 2 1
1421.2.b.b 2 609.h even 2 1
1856.2.e.f 2 24.f even 2 1
1856.2.e.f 2 696.l even 2 1
1856.2.e.g 2 24.h odd 2 1
1856.2.e.g 2 696.n odd 2 1
4176.2.o.k 2 4.b odd 2 1
4176.2.o.k 2 116.d odd 2 1
7569.2.a.i 2 29.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 45 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 20 \) Copy content Toggle raw display
$43$ \( T^{2} + 45 \) Copy content Toggle raw display
$47$ \( T^{2} + 5 \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 180 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 45 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 180 \) Copy content Toggle raw display
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