Properties

Label 6171.2.a.bb
Level $6171$
Weight $2$
Character orbit 6171.a
Self dual yes
Analytic conductor $49.276$
Analytic rank $0$
Dimension $6$
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] = [6171,2,Mod(1,6171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6171, 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("6171.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6171.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: \(49.2756830873\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + \beta_1 q^{6} - \beta_{4} q^{7} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + \beta_1 q^{6} - \beta_{4} q^{7} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{10} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{12} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{14} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{15} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{16} + q^{17} + \beta_1 q^{18} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{19} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{20} - \beta_{4} q^{21} + ( - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{23} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{24} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{25} + (\beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{26} + q^{27} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{28} + (\beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{30} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{32} + \beta_1 q^{34} + (\beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{36} + (2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{37} + (\beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{38} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{39} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 6) q^{40} + (2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{41} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{42} + ( - 4 \beta_{5} - 4 \beta_{2} - 2 \beta_1 - 3) q^{43} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{45} + ( - 2 \beta_{5} - 4 \beta_{4} - 4 \beta_{2} - 2 \beta_1 - 8) q^{46} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 5 \beta_{2} - \beta_1 - 4) q^{47} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{48} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{49} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2}) q^{50} + q^{51} + (4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{52} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 1) q^{53} + \beta_1 q^{54} + ( - \beta_{5} - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 5) q^{56} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{57} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{58} + ( - 2 \beta_{5} - 2 \beta_{4} - 2) q^{59} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{60} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{61} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{2} + \beta_1 - 8) q^{62} - \beta_{4} q^{63} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 4) q^{64} + (3 \beta_{5} + 2 \beta_{4} + \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 1) q^{65} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{67} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{68} + ( - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{69} + ( - \beta_{5} - 2 \beta_{3} - 5 \beta_{2} - \beta_1 - 6) q^{70} + ( - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{71} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{72} + (2 \beta_{5} - 2 \beta_{3} - 8 \beta_{2} - 4) q^{73} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} - 4 \beta_1) q^{74} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{75} + (3 \beta_{2} + 6) q^{76} + (\beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{78} + (2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2) q^{79} + (4 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 12) q^{80} + q^{81} + ( - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 6) q^{82} + (3 \beta_{5} + \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 4) q^{83} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{84} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{85} + ( - 2 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{86} + (\beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{87} + ( - 3 \beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_1 + 6) q^{89} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{90} + ( - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4) q^{91} + ( - 2 \beta_{4} - 8 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 6) q^{92} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + (\beta_{5} - 3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 4) q^{94} + ( - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{95} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{96} + ( - 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 4) q^{97} + ( - \beta_{5} + \beta_{2} - 4 \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 6 q^{3} + 5 q^{4} - 2 q^{5} + q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 6 q^{3} + 5 q^{4} - 2 q^{5} + q^{6} + 6 q^{8} + 6 q^{9} + 6 q^{10} + 5 q^{12} + 10 q^{13} - 6 q^{14} - 2 q^{15} + 7 q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + 8 q^{20} - 6 q^{23} + 6 q^{24} + 6 q^{25} + 7 q^{26} + 6 q^{27} - 20 q^{28} + 20 q^{29} + 6 q^{30} + 43 q^{32} + q^{34} - 2 q^{35} + 5 q^{36} - 6 q^{37} - 17 q^{38} + 10 q^{39} + 38 q^{40} + 16 q^{41} - 6 q^{42} - 2 q^{45} - 34 q^{46} - 10 q^{47} + 7 q^{48} - 14 q^{49} + 9 q^{50} + 6 q^{51} + 13 q^{52} + 14 q^{53} + q^{54} - 22 q^{56} - 4 q^{57} + 20 q^{58} - 8 q^{59} + 8 q^{60} + 22 q^{61} - 20 q^{62} + 28 q^{64} + 20 q^{65} + 16 q^{67} + 5 q^{68} - 6 q^{69} - 24 q^{70} - 18 q^{71} + 6 q^{72} - 8 q^{73} + 12 q^{74} + 6 q^{75} + 27 q^{76} + 7 q^{78} + 16 q^{79} + 66 q^{80} + 6 q^{81} - 24 q^{82} + 2 q^{83} - 20 q^{84} - 2 q^{85} - 17 q^{86} + 20 q^{87} + 32 q^{89} + 6 q^{90} + 26 q^{91} - 48 q^{92} - 51 q^{94} + 16 q^{95} + 43 q^{96} - 18 q^{97} + 19 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} - \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} + 6\nu^{3} - \nu^{2} - 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 5\nu^{3} - 7\nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 6\beta_{4} + 8\beta_{3} + 7\beta_{2} + 4\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 10\beta_{4} + 19\beta_{3} + 12\beta_{2} + 29\beta _1 + 16 \) Copy content Toggle raw display

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
−1.92474
−1.84175
−0.0970464
0.803425
1.32077
2.73934
−1.92474 1.00000 1.70460 −2.75408 −1.92474 −1.27612 0.568557 1.00000 5.30088
1.2 −1.84175 1.00000 1.39206 0.345659 −1.84175 0.969929 1.11968 1.00000 −0.636619
1.3 −0.0970464 1.00000 −1.99058 2.99589 −0.0970464 1.52957 0.387272 1.00000 −0.290740
1.4 0.803425 1.00000 −1.35451 −2.95862 0.803425 3.46909 −2.69510 1.00000 −2.37703
1.5 1.32077 1.00000 −0.255577 −2.10548 1.32077 −2.49950 −2.97909 1.00000 −2.78085
1.6 2.73934 1.00000 5.50401 2.47664 2.73934 −2.19296 9.59867 1.00000 6.78436
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(17\) \(-1\)

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 6171.2.a.bb yes 6
11.b odd 2 1 6171.2.a.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6171.2.a.y 6 11.b odd 2 1
6171.2.a.bb yes 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):

\( T_{2}^{6} - T_{2}^{5} - 8T_{2}^{4} + 5T_{2}^{3} + 14T_{2}^{2} - 9T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} - 16T_{5}^{4} - 30T_{5}^{3} + 68T_{5}^{2} + 108T_{5} - 44 \) Copy content Toggle raw display
\( T_{7}^{6} - 14T_{7}^{4} - 6T_{7}^{3} + 44T_{7}^{2} + 12T_{7} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} - 8 T^{4} + 5 T^{3} + 14 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} - 16 T^{4} - 30 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$7$ \( T^{6} - 14 T^{4} - 6 T^{3} + 44 T^{2} + \cdots - 36 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 10 T^{5} + 7 T^{4} + 208 T^{3} + \cdots - 99 \) Copy content Toggle raw display
$17$ \( (T - 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} - 31 T^{4} - 84 T^{3} + \cdots - 81 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} - 136 T^{4} + \cdots + 36416 \) Copy content Toggle raw display
$29$ \( T^{6} - 20 T^{5} + 56 T^{4} + \cdots - 25916 \) Copy content Toggle raw display
$31$ \( T^{6} - 86 T^{4} - 362 T^{3} + \cdots + 3916 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} - 128 T^{4} + \cdots + 28864 \) Copy content Toggle raw display
$41$ \( T^{6} - 16 T^{5} + 16 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{6} - 231 T^{4} - 272 T^{3} + \cdots - 193509 \) Copy content Toggle raw display
$47$ \( T^{6} + 10 T^{5} - 173 T^{4} + \cdots + 159031 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} - 116 T^{4} + \cdots - 304 \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{5} - 88 T^{4} + \cdots + 15616 \) Copy content Toggle raw display
$61$ \( T^{6} - 22 T^{5} + 154 T^{4} + \cdots + 524 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} - 147 T^{4} + \cdots + 22619 \) Copy content Toggle raw display
$71$ \( T^{6} + 18 T^{5} + 108 T^{4} + \cdots - 36 \) Copy content Toggle raw display
$73$ \( T^{6} + 8 T^{5} - 304 T^{4} + \cdots - 10496 \) Copy content Toggle raw display
$79$ \( T^{6} - 16 T^{5} - 44 T^{4} + \cdots - 6336 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} - 289 T^{4} + \cdots - 209 \) Copy content Toggle raw display
$89$ \( T^{6} - 32 T^{5} + 297 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} - 62 T^{4} + \cdots - 57364 \) Copy content Toggle raw display
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