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Properties

Label 6069.2.a.ba
Level $6069$
Weight $2$
Character orbit 6069.a
Self dual yes
Analytic conductor $48.461$
Analytic rank $0$
Dimension $9$
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] = [6069,2,Mod(1,6069)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, 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("6069.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.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: \(48.4612089867\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 51x^{3} - 27x^{2} + 6x + 1 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{5} - \beta_{4} + 1) q^{4} + (\beta_{6} + \beta_1) q^{5} - \beta_1 q^{6} + q^{7} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{5} - \beta_{4} + 1) q^{4} + (\beta_{6} + \beta_1) q^{5} - \beta_1 q^{6} + q^{7} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} + 6 q^{4} + 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} + 6 q^{4} + 3 q^{5} + 9 q^{7} + 9 q^{8} + 9 q^{9} + 12 q^{10} + 12 q^{11} - 6 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{19} + 15 q^{20} - 9 q^{21} + 6 q^{22} + 18 q^{23} - 9 q^{24} - 3 q^{26} - 9 q^{27} + 6 q^{28} + 6 q^{29} - 12 q^{30} + 30 q^{31} + 3 q^{32} - 12 q^{33} + 3 q^{35} + 6 q^{36} + 12 q^{37} + 24 q^{38} - 3 q^{39} + 30 q^{40} + 15 q^{41} - 12 q^{43} + 18 q^{44} + 3 q^{45} - 9 q^{46} + 15 q^{47} + 9 q^{49} - 12 q^{50} - 24 q^{52} - 6 q^{53} - 6 q^{55} + 9 q^{56} - 3 q^{57} + 3 q^{58} + 3 q^{59} - 15 q^{60} + 3 q^{61} + 30 q^{62} + 9 q^{63} - 27 q^{64} + 9 q^{65} - 6 q^{66} - 3 q^{67} - 18 q^{69} + 12 q^{70} + 12 q^{71} + 9 q^{72} + 15 q^{73} + 54 q^{74} + 3 q^{76} + 12 q^{77} + 3 q^{78} - 18 q^{79} + 9 q^{81} + 27 q^{82} + 24 q^{83} - 6 q^{84} - 60 q^{86} - 6 q^{87} + 36 q^{88} + 12 q^{90} + 3 q^{91} - 30 q^{93} - 36 q^{94} - 18 q^{95} - 3 q^{96} + 36 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 51x^{3} - 27x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 22\nu^{4} + 24\nu^{3} + 11\nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{8} - \nu^{7} - 10\nu^{6} + 7\nu^{5} + 30\nu^{4} - 9\nu^{3} - 27\nu^{2} - 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 22\nu^{4} - 31\nu^{3} - 12\nu^{2} + 12\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 22\nu^{4} - 31\nu^{3} - 11\nu^{2} + 12\nu - 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{8} - 5\nu^{7} - 28\nu^{6} + 38\nu^{5} + 74\nu^{4} - 63\nu^{3} - 50\nu^{2} + 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\nu^{8} - 4\nu^{7} - 30\nu^{6} + 29\nu^{5} + 89\nu^{4} - 41\nu^{3} - 74\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -3\nu^{8} + 5\nu^{7} + 29\nu^{6} - 38\nu^{5} - 82\nu^{4} + 63\nu^{3} + 64\nu^{2} - 10\nu - 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} + 5\beta_{5} - 6\beta_{4} + \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{6} + \beta_{5} - 7\beta_{4} - 7\beta_{3} + 8\beta_{2} + 26\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{8} - 8\beta_{7} + \beta_{6} + 26\beta_{5} - 34\beta_{4} + 8\beta_{3} + 10\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + 42\beta_{6} + 10\beta_{5} - 43\beta_{4} - 40\beta_{3} + 50\beta_{2} + 140\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -39\beta_{8} - 50\beta_{7} + 12\beta_{6} + 140\beta_{5} - 190\beta_{4} + 51\beta_{3} + 3\beta_{2} + 75\beta _1 + 387 \) 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
−2.29067
−1.59403
−1.56840
−0.765541
−0.118284
0.259117
1.30928
2.35958
2.40895
−2.29067 −1.00000 3.24715 −1.24679 2.29067 1.00000 −2.85680 1.00000 2.85597
1.2 −1.59403 −1.00000 0.540946 −3.07440 1.59403 1.00000 2.32578 1.00000 4.90069
1.3 −1.56840 −1.00000 0.459871 2.88639 1.56840 1.00000 2.41553 1.00000 −4.52700
1.4 −0.765541 −1.00000 −1.41395 2.98515 0.765541 1.00000 2.61352 1.00000 −2.28526
1.5 −0.118284 −1.00000 −1.98601 −1.64632 0.118284 1.00000 0.471482 1.00000 0.194734
1.6 0.259117 −1.00000 −1.93286 −1.75189 −0.259117 1.00000 −1.01907 1.00000 −0.453943
1.7 1.30928 −1.00000 −0.285783 0.212797 −1.30928 1.00000 −2.99273 1.00000 0.278612
1.8 2.35958 −1.00000 3.56759 2.62133 −2.35958 1.00000 3.69886 1.00000 6.18522
1.9 2.40895 −1.00000 3.80304 2.01373 −2.40895 1.00000 4.34343 1.00000 4.85097
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)
\(17\) \( +1 \)

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 6069.2.a.ba ✓ 9
17.b even 2 1 6069.2.a.bb yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6069.2.a.ba ✓ 9 1.a even 1 1 trivial
6069.2.a.bb yes 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):

\( T_{2}^{9} - 12T_{2}^{7} - 3T_{2}^{6} + 45T_{2}^{5} + 21T_{2}^{4} - 51T_{2}^{3} - 27T_{2}^{2} + 6T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{9} - 3T_{5}^{8} - 18T_{5}^{7} + 51T_{5}^{6} + 111T_{5}^{5} - 261T_{5}^{4} - 330T_{5}^{3} + 456T_{5}^{2} + 423T_{5} - 107 \) Copy content Toggle raw display
\( T_{11}^{9} - 12 T_{11}^{8} + 30 T_{11}^{7} + 147 T_{11}^{6} - 915 T_{11}^{5} + 1662 T_{11}^{4} + \cdots - 323 \) Copy content Toggle raw display
\( T_{23}^{9} - 18 T_{23}^{8} + 90 T_{23}^{7} + 90 T_{23}^{6} - 1890 T_{23}^{5} + 4266 T_{23}^{4} + \cdots + 7209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 12 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} + \cdots - 107 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - 12 T^{8} + \cdots - 323 \) Copy content Toggle raw display
$13$ \( T^{9} - 3 T^{8} + \cdots - 397 \) Copy content Toggle raw display
$17$ \( T^{9} \) Copy content Toggle raw display
$19$ \( T^{9} - 3 T^{8} + \cdots + 17 \) Copy content Toggle raw display
$23$ \( T^{9} - 18 T^{8} + \cdots + 7209 \) Copy content Toggle raw display
$29$ \( T^{9} - 6 T^{8} + \cdots - 60173 \) Copy content Toggle raw display
$31$ \( T^{9} - 30 T^{8} + \cdots - 194343 \) Copy content Toggle raw display
$37$ \( T^{9} - 12 T^{8} + \cdots - 161949 \) Copy content Toggle raw display
$41$ \( T^{9} - 15 T^{8} + \cdots + 241361 \) Copy content Toggle raw display
$43$ \( T^{9} + 12 T^{8} + \cdots - 86111 \) Copy content Toggle raw display
$47$ \( T^{9} - 15 T^{8} + \cdots + 4132731 \) Copy content Toggle raw display
$53$ \( T^{9} + 6 T^{8} + \cdots + 44067 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} + \cdots - 6140601 \) Copy content Toggle raw display
$61$ \( T^{9} - 3 T^{8} + \cdots + 507943 \) Copy content Toggle raw display
$67$ \( T^{9} + 3 T^{8} + \cdots - 161855839 \) Copy content Toggle raw display
$71$ \( T^{9} - 12 T^{8} + \cdots - 23362077 \) Copy content Toggle raw display
$73$ \( T^{9} - 15 T^{8} + \cdots - 765216021 \) Copy content Toggle raw display
$79$ \( T^{9} + 18 T^{8} + \cdots + 13722047 \) Copy content Toggle raw display
$83$ \( T^{9} - 24 T^{8} + \cdots - 125117463 \) Copy content Toggle raw display
$89$ \( T^{9} - 252 T^{7} + \cdots + 9673361 \) Copy content Toggle raw display
$97$ \( T^{9} - 36 T^{8} + \cdots + 6040389 \) Copy content Toggle raw display
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