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Properties

Label 5166.2.a.ca
Level $5166$
Weight $2$
Character orbit 5166.a
Self dual yes
Analytic conductor $41.251$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5166,2,Mod(1,5166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5166, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5166.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5166 = 2 \cdot 3^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5166.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: \(41.2507176842\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 21x^{5} + 18x^{4} + 150x^{3} + 8x^{2} - 358x - 260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_1 q^{5} - q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_1 q^{5} - q^{7} - q^{8} - \beta_1 q^{10} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{11} + (\beta_{6} + \beta_{3}) q^{13} + q^{14} + q^{16} + (\beta_{5} + \beta_1) q^{17} + (\beta_{5} - \beta_{4} + 1) q^{19} + \beta_1 q^{20} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{22} + (\beta_{5} - \beta_{4} + \beta_{2} + \cdots - 1) q^{23}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} + 7 q^{4} + 2 q^{5} - 7 q^{7} - 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} + 7 q^{4} + 2 q^{5} - 7 q^{7} - 7 q^{8} - 2 q^{10} - 5 q^{11} + 7 q^{14} + 7 q^{16} + 3 q^{17} + 6 q^{19} + 2 q^{20} + 5 q^{22} - 4 q^{23} + 11 q^{25} - 7 q^{28} - 15 q^{29} + 6 q^{31} - 7 q^{32} - 3 q^{34} - 2 q^{35} + 15 q^{37} - 6 q^{38} - 2 q^{40} + 7 q^{41} + 11 q^{43} - 5 q^{44} + 4 q^{46} - 18 q^{47} + 7 q^{49} - 11 q^{50} - 15 q^{53} + 18 q^{55} + 7 q^{56} + 15 q^{58} - 12 q^{59} + 5 q^{61} - 6 q^{62} + 7 q^{64} - 12 q^{65} + 14 q^{67} + 3 q^{68} + 2 q^{70} - 17 q^{71} + 17 q^{73} - 15 q^{74} + 6 q^{76} + 5 q^{77} + 22 q^{79} + 2 q^{80} - 7 q^{82} - 13 q^{83} + 39 q^{85} - 11 q^{86} + 5 q^{88} + 17 q^{89} - 4 q^{92} + 18 q^{94} - 19 q^{95} + 8 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 21x^{5} + 18x^{4} + 150x^{3} + 8x^{2} - 358x - 260 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 3\nu^{3} - 11\nu^{2} + 17\nu + 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 11\nu^{4} + 36\nu^{3} + 48\nu^{2} - 74\nu - 92 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 5\nu^{5} - 8\nu^{4} + 48\nu^{3} + 28\nu^{2} - 112\nu - 84 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 4\nu^{5} - 13\nu^{4} + 43\nu^{3} + 67\nu^{2} - 115\nu - 146 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{4} + 2\beta_{3} + 3\beta_{2} + 10\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{6} - 6\beta_{4} + 7\beta_{3} + 20\beta_{2} + 24\beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{6} - 2\beta_{5} - 40\beta_{4} + 45\beta_{3} + 76\beta_{2} + 134\beta _1 + 158 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 81\beta_{6} - 8\beta_{5} - 152\beta_{4} + 185\beta_{3} + 368\beta_{2} + 466\beta _1 + 786 \) Copy content Toggle raw display

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

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
−2.41359
−2.33220
−1.48004
−1.13397
2.38857
2.69352
4.27770
−1.00000 0 1.00000 −2.41359 0 −1.00000 −1.00000 0 2.41359
1.2 −1.00000 0 1.00000 −2.33220 0 −1.00000 −1.00000 0 2.33220
1.3 −1.00000 0 1.00000 −1.48004 0 −1.00000 −1.00000 0 1.48004
1.4 −1.00000 0 1.00000 −1.13397 0 −1.00000 −1.00000 0 1.13397
1.5 −1.00000 0 1.00000 2.38857 0 −1.00000 −1.00000 0 −2.38857
1.6 −1.00000 0 1.00000 2.69352 0 −1.00000 −1.00000 0 −2.69352
1.7 −1.00000 0 1.00000 4.27770 0 −1.00000 −1.00000 0 −4.27770
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(1\)
\(41\) \(-1\)

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 5166.2.a.ca ✓ 7
3.b odd 2 1 5166.2.a.cb yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5166.2.a.ca ✓ 7 1.a even 1 1 trivial
5166.2.a.cb yes 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5166))\):

\( T_{5}^{7} - 2T_{5}^{6} - 21T_{5}^{5} + 18T_{5}^{4} + 150T_{5}^{3} + 8T_{5}^{2} - 358T_{5} - 260 \) Copy content Toggle raw display
\( T_{11}^{7} + 5T_{11}^{6} - 60T_{11}^{5} - 348T_{11}^{4} + 546T_{11}^{3} + 5910T_{11}^{2} + 10568T_{11} + 5696 \) Copy content Toggle raw display
\( T_{13}^{7} - 71T_{13}^{5} + 110T_{13}^{4} + 1240T_{13}^{3} - 3888T_{13}^{2} + 3312T_{13} - 864 \) Copy content Toggle raw display
\( T_{17}^{7} - 3T_{17}^{6} - 60T_{17}^{5} + 244T_{17}^{4} + 372T_{17}^{3} - 2988T_{17}^{2} + 4544T_{17} - 2048 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 2 T^{6} + \cdots - 260 \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 5 T^{6} + \cdots + 5696 \) Copy content Toggle raw display
$13$ \( T^{7} - 71 T^{5} + \cdots - 864 \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} + \cdots - 2048 \) Copy content Toggle raw display
$19$ \( T^{7} - 6 T^{6} + \cdots + 640 \) Copy content Toggle raw display
$23$ \( T^{7} + 4 T^{6} + \cdots - 17392 \) Copy content Toggle raw display
$29$ \( T^{7} + 15 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{7} - 6 T^{6} + \cdots + 640 \) Copy content Toggle raw display
$37$ \( T^{7} - 15 T^{6} + \cdots - 2560 \) Copy content Toggle raw display
$41$ \( (T - 1)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} - 11 T^{6} + \cdots + 372608 \) Copy content Toggle raw display
$47$ \( T^{7} + 18 T^{6} + \cdots - 83968 \) Copy content Toggle raw display
$53$ \( T^{7} + 15 T^{6} + \cdots - 1304 \) Copy content Toggle raw display
$59$ \( T^{7} + 12 T^{6} + \cdots - 311936 \) Copy content Toggle raw display
$61$ \( T^{7} - 5 T^{6} + \cdots - 1504 \) Copy content Toggle raw display
$67$ \( T^{7} - 14 T^{6} + \cdots - 548068 \) Copy content Toggle raw display
$71$ \( T^{7} + 17 T^{6} + \cdots + 27136 \) Copy content Toggle raw display
$73$ \( T^{7} - 17 T^{6} + \cdots + 530368 \) Copy content Toggle raw display
$79$ \( T^{7} - 22 T^{6} + \cdots + 10240 \) Copy content Toggle raw display
$83$ \( T^{7} + 13 T^{6} + \cdots + 32240 \) Copy content Toggle raw display
$89$ \( T^{7} - 17 T^{6} + \cdots + 4345856 \) Copy content Toggle raw display
$97$ \( T^{7} - 8 T^{6} + \cdots - 192512 \) Copy content Toggle raw display
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