Properties

Label 261.2.g.b
Level $261$
Weight $2$
Character orbit 261.g
Analytic conductor $2.084$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [261,2,Mod(17,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.g (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} + 2 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} - q^{7} + (\beta_{7} - 3 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{6} + 2 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} - q^{7} + (\beta_{7} - 3 \beta_{3}) q^{8} + ( - \beta_{2} - 1) q^{10} + (\beta_{4} - \beta_1) q^{11} - 5 \beta_{2} q^{13} + \beta_1 q^{14} + (\beta_{5} - 7) q^{16} + (\beta_{4} - 3 \beta_1) q^{17} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2} + 2) q^{19} + ( - 2 \beta_{7} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{20} + (\beta_{6} + 3 \beta_{2}) q^{22} + (\beta_{3} + \beta_1) q^{23} + (2 \beta_{5} + 1) q^{25} + 5 \beta_{3} q^{26} + ( - \beta_{6} - 2 \beta_{2}) q^{28} + ( - \beta_{7} - 2 \beta_{4} - \beta_{3} + 2 \beta_1) q^{29} + (\beta_{6} + \beta_{5} - \beta_{2} + 1) q^{31} + (\beta_{4} + 4 \beta_1) q^{32} + (3 \beta_{6} + 11 \beta_{2}) q^{34} + (\beta_{7} - \beta_{4}) q^{35} + ( - \beta_{6} + \beta_{5} + \beta_{2} + 1) q^{37} + ( - \beta_{7} + \beta_{4} + 5 \beta_{3} - 5 \beta_1) q^{38} + ( - \beta_{6} - \beta_{5}) q^{40} + (2 \beta_{7} + 2 \beta_{3}) q^{41} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{43} + ( - \beta_{7} - 4 \beta_{3}) q^{44} + ( - \beta_{6} - \beta_{5} - 4 \beta_{2} + 4) q^{46} + (5 \beta_{7} - 3 \beta_{3}) q^{47} - 6 q^{49} + ( - 2 \beta_{4} + 5 \beta_1) q^{50} + ( - 5 \beta_{5} + 10) q^{52} + (\beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - \beta_{6} + \beta_{5} + 2 \beta_{2} + 2) q^{55} + ( - \beta_{7} + 3 \beta_{3}) q^{56} + ( - 2 \beta_{6} + \beta_{5} - 6 \beta_{2} - 5) q^{58} + (3 \beta_{7} + 3 \beta_{4} - \beta_{3} - \beta_1) q^{59} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{61} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{62} + ( - 2 \beta_{6} - 3 \beta_{2}) q^{64} + ( - 5 \beta_{7} - 5 \beta_{4}) q^{65} + ( - 2 \beta_{6} + 7 \beta_{2}) q^{67} + (\beta_{7} - 14 \beta_{3}) q^{68} + (\beta_{2} + 1) q^{70} + ( - 3 \beta_{3} + 3 \beta_1) q^{71} + (\beta_{6} - \beta_{5} - 5 \beta_{2} - 5) q^{73} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{74} + (3 \beta_{6} - 3 \beta_{5} + 15 \beta_{2} + 15) q^{76} + ( - \beta_{4} + \beta_1) q^{77} + (\beta_{6} + \beta_{5} - 6 \beta_{2} + 6) q^{79} + (3 \beta_{7} - 3 \beta_{4} + \beta_{3} - \beta_1) q^{80} + ( - 2 \beta_{5} + 10) q^{82} + ( - 3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{83} + ( - \beta_{6} + \beta_{5}) q^{85} + ( - 2 \beta_{7} + 2 \beta_{4} + 8 \beta_{3} - 8 \beta_1) q^{86} + (2 \beta_{5} - 11) q^{88} + (7 \beta_{4} - \beta_1) q^{89} + 5 \beta_{2} q^{91} + ( - \beta_{7} + \beta_{4} + 5 \beta_{3} - 5 \beta_1) q^{92} + (3 \beta_{5} - 7) q^{94} + (4 \beta_{7} - 2 \beta_{3}) q^{95} + (\beta_{6} - \beta_{5} + 8 \beta_{2} + 8) q^{97} + 6 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 8 q^{10} - 52 q^{16} + 12 q^{19} + 16 q^{25} + 12 q^{31} + 12 q^{37} - 4 q^{40} + 8 q^{43} + 28 q^{46} - 48 q^{49} + 60 q^{52} + 20 q^{55} - 36 q^{58} - 4 q^{61} + 8 q^{70} - 44 q^{73} + 108 q^{76} + 52 q^{79} + 72 q^{82} + 4 q^{85} - 80 q^{88} - 44 q^{94} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 5\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} + 13\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 31\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 34\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 5\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{4} + 13\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{6} - 31\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{7} + 34\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(118\) \(146\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.14412 1.14412i
0.437016 + 0.437016i
−0.437016 0.437016i
1.14412 + 1.14412i
−1.14412 + 1.14412i
0.437016 0.437016i
−0.437016 + 0.437016i
1.14412 1.14412i
−1.85123 1.85123i 0 4.85410i 0.540182 0 −1.00000 5.28360 5.28360i 0 −1.00000 1.00000i
17.2 −0.270091 0.270091i 0 1.85410i 3.70246 0 −1.00000 −1.04096 + 1.04096i 0 −1.00000 1.00000i
17.3 0.270091 + 0.270091i 0 1.85410i −3.70246 0 −1.00000 1.04096 1.04096i 0 −1.00000 1.00000i
17.4 1.85123 + 1.85123i 0 4.85410i −0.540182 0 −1.00000 −5.28360 + 5.28360i 0 −1.00000 1.00000i
215.1 −1.85123 + 1.85123i 0 4.85410i 0.540182 0 −1.00000 5.28360 + 5.28360i 0 −1.00000 + 1.00000i
215.2 −0.270091 + 0.270091i 0 1.85410i 3.70246 0 −1.00000 −1.04096 1.04096i 0 −1.00000 + 1.00000i
215.3 0.270091 0.270091i 0 1.85410i −3.70246 0 −1.00000 1.04096 + 1.04096i 0 −1.00000 + 1.00000i
215.4 1.85123 1.85123i 0 4.85410i −0.540182 0 −1.00000 −5.28360 5.28360i 0 −1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.c odd 4 1 inner
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.g.b 8
3.b odd 2 1 inner 261.2.g.b 8
29.c odd 4 1 inner 261.2.g.b 8
87.f even 4 1 inner 261.2.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.g.b 8 1.a even 1 1 trivial
261.2.g.b 8 3.b odd 2 1 inner
261.2.g.b 8 29.c odd 4 1 inner
261.2.g.b 8 87.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 47T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 47T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 14 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 3122 T^{4} + 14641 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 14 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 1198 T^{4} + 707281 \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + 18 T^{2} + 108 T + 324)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + 8 T^{2} + 352 T + 7744)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 21602 T^{4} + \cdots + 25411681 \) Copy content Toggle raw display
$53$ \( (T^{4} + 54 T^{2} + 324)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 116 T^{2} + 484)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + 2 T^{2} - 44 T + 484)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 218 T^{2} + 361)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 126 T^{2} + 324)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 22 T^{3} + 242 T^{2} + 836 T + 1444)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 26 T^{3} + 338 T^{2} - 1612 T + 3844)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} + 103682T^{4} + 1 \) Copy content Toggle raw display
$97$ \( (T^{4} - 30 T^{3} + 450 T^{2} - 2700 T + 8100)^{2} \) Copy content Toggle raw display
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