Properties

Label 9036.2.a.j
Level $9036$
Weight $2$
Character orbit 9036.a
Self dual yes
Analytic conductor $72.153$
Analytic rank $1$
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] = [9036,2,Mod(1,9036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9036, 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("9036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9036 = 2^{2} \cdot 3^{2} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9036.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: \(72.1528232664\)
Analytic rank: \(1\)
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: \( x^{9} - 4x^{8} - 8x^{7} + 36x^{6} + 19x^{5} - 110x^{4} - 13x^{3} + 136x^{2} - x - 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3012)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - \beta_{7} q^{7} + ( - \beta_{7} - \beta_{4} - \beta_1 - 1) q^{11} + (\beta_{8} - 1) q^{13} + (\beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_1 + 2) q^{17} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{23} + ( - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_1 - 1) q^{25} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{29} + ( - 2 \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{35} + (2 \beta_{7} + 3 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{8} + \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{43} + ( - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{47} + (2 \beta_{4} + 3 \beta_{3} - 1) q^{49} + ( - \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{53}+ \cdots + ( - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{5} - 5 q^{11} - 11 q^{13} + 9 q^{17} - 7 q^{19} - 10 q^{23} - 2 q^{25} + 9 q^{29} - 2 q^{31} + 3 q^{35} - 10 q^{37} + 5 q^{41} - 17 q^{43} - 11 q^{47} - 5 q^{49} + 12 q^{53} - 4 q^{55} - 13 q^{59} - 23 q^{61} + 4 q^{65} - 7 q^{67} - 15 q^{71} - 35 q^{73} + 26 q^{77} - 18 q^{79} + 15 q^{83} - 29 q^{85} + 8 q^{89} - 20 q^{91} - 9 q^{95} - 43 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 8x^{7} + 36x^{6} + 19x^{5} - 110x^{4} - 13x^{3} + 136x^{2} - x - 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 9\nu^{5} + 20\nu^{4} + 24\nu^{3} - 39\nu^{2} - 16\nu + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} - 3\nu^{7} - 9\nu^{6} + 20\nu^{5} + 24\nu^{4} - 37\nu^{3} - 22\nu^{2} + 16\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 4\nu^{7} - 6\nu^{6} + 29\nu^{5} + 4\nu^{4} - 61\nu^{3} + 15\nu^{2} + 38\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 3\nu^{7} - 12\nu^{6} + 27\nu^{5} + 55\nu^{4} - 75\nu^{3} - 112\nu^{2} + 61\nu + 80 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 3\nu^{7} - 10\nu^{6} + 22\nu^{5} + 35\nu^{4} - 47\nu^{3} - 53\nu^{2} + 29\nu + 29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} - 6\nu^{7} - 21\nu^{6} + 47\nu^{5} + 79\nu^{4} - 112\nu^{3} - 134\nu^{2} + 81\nu + 86 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{8} + 7\nu^{7} + 39\nu^{6} - 56\nu^{5} - 174\nu^{4} + 141\nu^{3} + 316\nu^{2} - 112\nu - 194 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{8} + 11\nu^{7} + 46\nu^{6} - 88\nu^{5} - 185\nu^{4} + 216\nu^{3} + 311\nu^{2} - 161\nu - 184 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - 3\beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 15\beta_{6} - 11\beta_{4} - 8\beta_{3} - 5\beta_{2} - 4\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + 53\beta_{6} - 2\beta_{5} - 43\beta_{4} - 32\beta_{3} - 13\beta_{2} - 26\beta _1 + 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{8} + 20 \beta_{7} + 211 \beta_{6} - 10 \beta_{5} - 155 \beta_{4} - 122 \beta_{3} - 49 \beta_{2} - 78 \beta _1 + 169 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 30 \beta_{8} + 42 \beta_{7} + 775 \beta_{6} - 44 \beta_{5} - 593 \beta_{4} - 454 \beta_{3} - 169 \beta_{2} - 340 \beta _1 + 651 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 86 \beta_{8} + 218 \beta_{7} + 2937 \beta_{6} - 182 \beta_{5} - 2183 \beta_{4} - 1706 \beta_{3} - 623 \beta_{2} - 1180 \beta _1 + 2307 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 400 \beta_{8} + 658 \beta_{7} + 10899 \beta_{6} - 694 \beta_{5} - 8211 \beta_{4} - 6336 \beta_{3} - 2285 \beta_{2} - 4608 \beta _1 + 8683 ) / 2 \) 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
1.62442
−1.19695
1.92557
−0.900134
3.72676
−2.04788
1.14273
1.27419
−1.54872
0 0 0 −2.54312 0 1.31259 0 0 0
1.2 0 0 0 −2.10983 0 −4.41753 0 0 0
1.3 0 0 0 −1.12750 0 −1.09363 0 0 0
1.4 0 0 0 −0.756569 0 2.51040 0 0 0
1.5 0 0 0 −0.313469 0 −0.749031 0 0 0
1.6 0 0 0 0.933484 0 1.18850 0 0 0
1.7 0 0 0 1.64620 0 4.41772 0 0 0
1.8 0 0 0 3.38480 0 −2.75780 0 0 0
1.9 0 0 0 3.88600 0 −0.411221 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(251\) \(1\)

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 9036.2.a.j 9
3.b odd 2 1 3012.2.a.g 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3012.2.a.g 9 3.b odd 2 1
9036.2.a.j 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9036))\):

\( T_{5}^{9} - 3T_{5}^{8} - 17T_{5}^{7} + 35T_{5}^{6} + 105T_{5}^{5} - 89T_{5}^{4} - 216T_{5}^{3} + 22T_{5}^{2} + 117T_{5} + 29 \) Copy content Toggle raw display
\( T_{17}^{9} - 9 T_{17}^{8} - 36 T_{17}^{7} + 394 T_{17}^{6} + 531 T_{17}^{5} - 5615 T_{17}^{4} - 5263 T_{17}^{3} + 24899 T_{17}^{2} + 23273 T_{17} - 8179 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} - 17 T^{7} + 35 T^{6} + \cdots + 29 \) Copy content Toggle raw display
$7$ \( T^{9} - 29 T^{7} + T^{6} + 204 T^{5} + \cdots + 71 \) Copy content Toggle raw display
$11$ \( T^{9} + 5 T^{8} - 20 T^{7} - 151 T^{6} + \cdots - 288 \) Copy content Toggle raw display
$13$ \( T^{9} + 11 T^{8} + 28 T^{7} + \cdots - 688 \) Copy content Toggle raw display
$17$ \( T^{9} - 9 T^{8} - 36 T^{7} + \cdots - 8179 \) Copy content Toggle raw display
$19$ \( T^{9} + 7 T^{8} - 69 T^{7} + \cdots + 74304 \) Copy content Toggle raw display
$23$ \( T^{9} + 10 T^{8} - 73 T^{7} + \cdots - 150037 \) Copy content Toggle raw display
$29$ \( T^{9} - 9 T^{8} - 99 T^{7} + \cdots + 43088 \) Copy content Toggle raw display
$31$ \( T^{9} + 2 T^{8} - 137 T^{7} + \cdots - 81733 \) Copy content Toggle raw display
$37$ \( T^{9} + 10 T^{8} - 147 T^{7} + \cdots - 284092 \) Copy content Toggle raw display
$41$ \( T^{9} - 5 T^{8} - 232 T^{7} + \cdots + 155867 \) Copy content Toggle raw display
$43$ \( T^{9} + 17 T^{8} - 91 T^{7} + \cdots + 16200 \) Copy content Toggle raw display
$47$ \( T^{9} + 11 T^{8} - 129 T^{7} + \cdots - 104672 \) Copy content Toggle raw display
$53$ \( T^{9} - 12 T^{8} - 264 T^{7} + \cdots - 2347892 \) Copy content Toggle raw display
$59$ \( T^{9} + 13 T^{8} - 209 T^{7} + \cdots + 17591076 \) Copy content Toggle raw display
$61$ \( T^{9} + 23 T^{8} + 90 T^{7} + \cdots - 1398596 \) Copy content Toggle raw display
$67$ \( T^{9} + 7 T^{8} - 91 T^{7} + \cdots - 17383 \) Copy content Toggle raw display
$71$ \( T^{9} + 15 T^{8} - 282 T^{7} + \cdots - 52793236 \) Copy content Toggle raw display
$73$ \( T^{9} + 35 T^{8} + 275 T^{7} + \cdots + 15849671 \) Copy content Toggle raw display
$79$ \( T^{9} + 18 T^{8} - 310 T^{7} + \cdots + 83030011 \) Copy content Toggle raw display
$83$ \( T^{9} - 15 T^{8} - 334 T^{7} + \cdots - 2137273 \) Copy content Toggle raw display
$89$ \( T^{9} - 8 T^{8} - 359 T^{7} + \cdots + 17515072 \) Copy content Toggle raw display
$97$ \( T^{9} + 43 T^{8} + 424 T^{7} + \cdots - 10479024 \) Copy content Toggle raw display
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