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Properties

Label 261.2.c.c
Level $261$
Weight $2$
Character orbit 261.c
Analytic conductor $2.084$
Analytic rank $0$
Dimension $6$
CM discriminant -87
Inner twists $4$

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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: \(6\)
Coefficient field: 6.0.10241038656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + ( - \beta_{4} + \beta_{2}) q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + ( - \beta_{4} + \beta_{2}) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + \beta_{5} q^{11} + (\beta_{4} + \beta_{2}) q^{13} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{14} + (\beta_{4} - 4 \beta_{2} + 4) q^{16} + ( - \beta_{5} + 2 \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{22} - 5 q^{25} + (\beta_{5} + \beta_{3} - 3 \beta_1) q^{26} + (\beta_{4} - 4 \beta_{2} + 5) q^{28} - \beta_{3} q^{29} + (\beta_{5} - 2 \beta_{3} + 7 \beta_1) q^{32} + (2 \beta_{4} + \beta_{2} - 7) q^{34} - 2 \beta_{3} q^{41} + (\beta_{3} - \beta_1) q^{44} + ( - \beta_{5} - 4 \beta_1) q^{47} + ( - 3 \beta_{4} + \beta_{2} + 7) q^{49} - 5 \beta_1 q^{50} + (\beta_{4} - 4 \beta_{2} + 11) q^{52} + ( - \beta_{5} - 2 \beta_{3} + 10 \beta_1) q^{56} + ( - \beta_{4} + 4 \beta_{2}) q^{58} + ( - 2 \beta_{4} + 8 \beta_{2} - 21) q^{64} + (3 \beta_{4} + \beta_{2}) q^{67} + (\beta_{3} - 7 \beta_1) q^{68} + (\beta_{5} - 2 \beta_{3} - 6 \beta_1) q^{77} + ( - 2 \beta_{4} + 8 \beta_{2}) q^{82} + ( - 3 \beta_{4} - 3 \beta_{2} + 2) q^{88} + (\beta_{5} + 6 \beta_1) q^{89} + (\beta_{4} - 5 \beta_{2} - 4) q^{91} + (2 \beta_{4} - 5 \beta_{2} + 17) q^{94} + ( - 3 \beta_{5} + \beta_{3} + 8 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{4} + 24 q^{16} - 6 q^{22} - 30 q^{25} + 30 q^{28} - 42 q^{34} + 42 q^{49} + 66 q^{52} - 126 q^{64} + 12 q^{88} - 24 q^{91} + 102 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 10\nu^{2} + 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 10\nu^{3} + 17\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 10\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 10\beta_{3} + 43\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 2.81322i
− 1.66024i
− 1.15299i
1.15299i
1.66024i
2.81322i
− 2.81322i 0 −5.91423 0 0 −3.40698 11.0116i 0 0
28.2 − 1.66024i 0 −0.756389 0 0 5.20982 − 2.06469i 0 0
28.3 − 1.15299i 0 0.670622 0 0 −1.80284 − 3.07919i 0 0
28.4 1.15299i 0 0.670622 0 0 −1.80284 3.07919i 0 0
28.5 1.66024i 0 −0.756389 0 0 5.20982 2.06469i 0 0
28.6 2.81322i 0 −5.91423 0 0 −3.40698 − 11.0116i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)
3.b odd 2 1 inner
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.c ✓ 6
3.b odd 2 1 inner 261.2.c.c ✓ 6
4.b odd 2 1 4176.2.o.p 6
12.b even 2 1 4176.2.o.p 6
29.b even 2 1 inner 261.2.c.c ✓ 6
29.c odd 4 2 7569.2.a.ba 6
87.d odd 2 1 CM 261.2.c.c ✓ 6
87.f even 4 2 7569.2.a.ba 6
116.d odd 2 1 4176.2.o.p 6
348.b even 2 1 4176.2.o.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.c.c ✓ 6 1.a even 1 1 trivial
261.2.c.c ✓ 6 3.b odd 2 1 inner
261.2.c.c ✓ 6 29.b even 2 1 inner
261.2.c.c ✓ 6 87.d odd 2 1 CM
4176.2.o.p 6 4.b odd 2 1
4176.2.o.p 6 12.b even 2 1
4176.2.o.p 6 116.d odd 2 1
4176.2.o.p 6 348.b even 2 1
7569.2.a.ba 6 29.c odd 4 2
7569.2.a.ba 6 87.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 12T_{2}^{4} + 36T_{2}^{2} + 29 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 12 T^{4} + \cdots + 29 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 21 T - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 66 T^{4} + \cdots + 1856 \) Copy content Toggle raw display
$13$ \( (T^{3} - 39 T - 86)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 102 T^{4} + \cdots + 19604 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( (T^{2} + 29)^{3} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( (T^{2} + 116)^{3} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} + 282 T^{4} + \cdots + 56144 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( (T^{3} - 201 T - 932)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} + 534 T^{4} + \cdots + 1481204 \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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