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Properties

Label 9025.2.a.cc
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9025,2,Mod(1,9025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-6,-9,6,0,12,0,-6,6,0,0,-18,-9,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{6} + \beta_{4}) q^{4} + ( - \beta_{7} + \beta_{3} - \beta_1 + 2) q^{6} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{7} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots + \beta_1) q^{8}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} - 9 q^{3} + 6 q^{4} + 12 q^{6} - 6 q^{8} + 6 q^{9} - 18 q^{12} - 9 q^{13} + 12 q^{16} + 9 q^{17} - 24 q^{18} + 12 q^{21} - 24 q^{22} + 12 q^{23} + 3 q^{24} - 3 q^{26} - 24 q^{27} + 15 q^{28}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 10\nu^{3} + 4\nu^{2} - 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 3\nu^{6} - 4\nu^{5} + 10\nu^{4} + 4\nu^{3} - 7\nu^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 7\nu^{4} - 14\nu^{3} - \nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 13\nu^{4} + 7\nu^{3} - 14\nu^{2} - 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 5\nu^{6} - 13\nu^{5} - 7\nu^{4} + 14\nu^{3} + 2\nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 8\nu^{4} - 17\nu^{3} - 4\nu^{2} + 8\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} - 7\nu^{7} - 7\nu^{6} + 30\nu^{5} + 5\nu^{4} - 34\nu^{3} - \nu^{2} + 7\nu + 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + 3\beta_{6} + 2\beta_{4} + \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} - 2\beta_{7} + 12\beta_{6} + 8\beta_{4} + 3\beta_{3} + 23\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{8} - 9\beta_{7} + 38\beta_{6} - \beta_{5} + 23\beta_{4} + 13\beta_{3} + 77\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 38\beta_{8} - 25\beta_{7} + 128\beta_{6} - 3\beta_{5} + 77\beta_{4} + 41\beta_{3} + \beta_{2} + 242\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 128\beta_{8} - 87\beta_{7} + 411\beta_{6} - 13\beta_{5} + 242\beta_{4} + 142\beta_{3} + 3\beta_{2} + 786\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 411 \beta_{8} - 269 \beta_{7} + 1338 \beta_{6} - 41 \beta_{5} + 786 \beta_{4} + 455 \beta_{3} + \cdots + 46 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.58312
−1.28997
−1.13237
−0.256961
0.309891
0.789016
1.30799
1.63278
3.22274
−2.58312 0.247720 4.67249 0 −0.639888 0.401640 −6.90335 −2.93864 0
1.2 −2.28997 −3.30730 3.24395 0 7.57362 2.93910 −2.84861 7.93825 0
1.3 −2.13237 −2.23040 2.54700 0 4.75604 −1.48562 −1.16642 1.97468 0
1.4 −1.25696 −3.01225 −0.420048 0 3.78628 −3.72392 3.04191 6.07366 0
1.5 −0.690109 0.694850 −1.52375 0 −0.479522 2.33464 2.43177 −2.51718 0
1.6 −0.210984 −0.0798955 −1.95549 0 0.0168566 1.68723 0.834543 −2.99362 0
1.7 0.307988 1.64392 −1.90514 0 0.506308 −0.0891959 −1.20274 −0.297533 0
1.8 0.632780 −1.91964 −1.59959 0 −1.21471 −4.08895 −2.27775 0.685005 0
1.9 2.22274 −1.03700 2.94057 0 −2.30498 2.02508 2.09064 −1.92463 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
\(5\) \( +1 \)
\(19\) \( +1 \)

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 9025.2.a.cc 9
5.b even 2 1 1805.2.a.v 9
19.b odd 2 1 9025.2.a.cf 9
19.e even 9 2 475.2.l.c 18
95.d odd 2 1 1805.2.a.s 9
95.p even 18 2 95.2.k.a ✓ 18
95.q odd 36 4 475.2.u.b 36
285.bd odd 18 2 855.2.bs.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a ✓ 18 95.p even 18 2
475.2.l.c 18 19.e even 9 2
475.2.u.b 36 95.q odd 36 4
855.2.bs.c 18 285.bd odd 18 2
1805.2.a.s 9 95.d odd 2 1
1805.2.a.v 9 5.b even 2 1
9025.2.a.cc 9 1.a even 1 1 trivial
9025.2.a.cf 9 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{9} + 6T_{2}^{8} + 6T_{2}^{7} - 26T_{2}^{6} - 60T_{2}^{5} - 21T_{2}^{4} + 30T_{2}^{3} + 15T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{9} + 9T_{3}^{8} + 24T_{3}^{7} - T_{3}^{6} - 87T_{3}^{5} - 81T_{3}^{4} + 44T_{3}^{3} + 48T_{3}^{2} - 9T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{9} - 27T_{7}^{7} + 24T_{7}^{6} + 213T_{7}^{5} - 342T_{7}^{4} - 277T_{7}^{3} + 651T_{7}^{2} - 153T_{7} - 19 \) Copy content Toggle raw display
\( T_{11}^{9} - 60T_{11}^{7} - 52T_{11}^{6} + 1218T_{11}^{5} + 2037T_{11}^{4} - 7981T_{11}^{3} - 20100T_{11}^{2} - 7353T_{11} + 773 \) Copy content Toggle raw display
\( T_{29}^{9} - 9 T_{29}^{8} - 66 T_{29}^{7} + 673 T_{29}^{6} + 1278 T_{29}^{5} - 16863 T_{29}^{4} + \cdots - 210403 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 6 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{9} + 9 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 27 T^{7} + \cdots - 19 \) Copy content Toggle raw display
$11$ \( T^{9} - 60 T^{7} + \cdots + 773 \) Copy content Toggle raw display
$13$ \( T^{9} + 9 T^{8} + \cdots - 53 \) Copy content Toggle raw display
$17$ \( T^{9} - 9 T^{8} + \cdots - 1691 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 12 T^{8} + \cdots + 45667 \) Copy content Toggle raw display
$29$ \( T^{9} - 9 T^{8} + \cdots - 210403 \) Copy content Toggle raw display
$31$ \( T^{9} - 18 T^{8} + \cdots + 216991 \) Copy content Toggle raw display
$37$ \( T^{9} + 18 T^{8} + \cdots + 11125 \) Copy content Toggle raw display
$41$ \( T^{9} - 6 T^{8} + \cdots + 361 \) Copy content Toggle raw display
$43$ \( T^{9} - 12 T^{8} + \cdots - 1591019 \) Copy content Toggle raw display
$47$ \( T^{9} + 15 T^{8} + \cdots - 1425943 \) Copy content Toggle raw display
$53$ \( T^{9} + 15 T^{8} + \cdots + 211859 \) Copy content Toggle raw display
$59$ \( T^{9} - 21 T^{8} + \cdots + 733771 \) Copy content Toggle raw display
$61$ \( T^{9} + 12 T^{8} + \cdots + 61561 \) Copy content Toggle raw display
$67$ \( T^{9} + 60 T^{8} + \cdots + 6493589 \) Copy content Toggle raw display
$71$ \( T^{9} + 18 T^{8} + \cdots - 53239843 \) Copy content Toggle raw display
$73$ \( T^{9} + 27 T^{8} + \cdots + 24197203 \) Copy content Toggle raw display
$79$ \( T^{9} - 15 T^{8} + \cdots + 3833 \) Copy content Toggle raw display
$83$ \( T^{9} - 507 T^{7} + \cdots - 189817057 \) Copy content Toggle raw display
$89$ \( T^{9} + 39 T^{8} + \cdots + 957419 \) Copy content Toggle raw display
$97$ \( T^{9} + 15 T^{8} + \cdots + 1914625 \) Copy content Toggle raw display
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