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Properties

Label 9075.2.a.dt
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $8$
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] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9} + ( - \beta_{2} - 1) q^{12} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{7} + \beta_{4} - 2 \beta_{3} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28} + 28 q^{29} + 10 q^{31} - 12 q^{32} - 8 q^{34} + 7 q^{36} - 14 q^{37} - 19 q^{38} + 7 q^{39} + 14 q^{41} + 8 q^{42} - 11 q^{43} - 5 q^{47} + 7 q^{48} + 2 q^{49} - q^{51} - q^{52} + 5 q^{53} + q^{54} - 4 q^{56} - 6 q^{57} + 3 q^{58} - 2 q^{59} + 36 q^{61} + 17 q^{62} - 8 q^{63} - 23 q^{64} - 6 q^{67} + 3 q^{68} + 7 q^{69} - 17 q^{71} - 3 q^{72} - 32 q^{73} + 47 q^{74} + 5 q^{76} + q^{78} + 41 q^{79} + 8 q^{81} + 6 q^{83} + 11 q^{84} - 9 q^{86} - 28 q^{87} + 21 q^{89} - 9 q^{91} - 69 q^{92} - 10 q^{93} + 51 q^{94} + 12 q^{96} + 4 q^{97} + 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 11\nu^{3} + 7\nu^{2} - 11\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 11\nu^{4} + 7\nu^{3} - 11\nu^{2} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 6\nu^{5} - 11\nu^{4} - 5\nu^{3} + 11\nu^{2} - 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 8\nu^{4} - 13\nu^{3} - 17\nu^{2} + 17\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 8\nu^{5} - 13\nu^{4} - 19\nu^{3} + 19\nu^{2} + 13\nu - 2 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + 2\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{6} + 7\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 26\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 8\beta_{6} + 9\beta_{5} + 11\beta_{4} + 10\beta_{3} + 25\beta_{2} + 20\beta _1 + 62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{7} + 11\beta_{6} + 42\beta_{5} + 54\beta_{4} + 15\beta_{3} + 12\beta_{2} + 140\beta _1 + 26 \) 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
2.49311
2.02189
1.40736
0.684625
−0.188059
−1.27571
−2.04884
−2.09438
−2.49311 −1.00000 4.21558 0 2.49311 −1.92071 −5.52367 1.00000 0
1.2 −2.02189 −1.00000 2.08806 0 2.02189 1.30998 −0.178044 1.00000 0
1.3 −1.40736 −1.00000 −0.0193259 0 1.40736 2.16377 2.84193 1.00000 0
1.4 −0.684625 −1.00000 −1.53129 0 0.684625 −4.22715 2.41761 1.00000 0
1.5 0.188059 −1.00000 −1.96463 0 −0.188059 −1.64886 −0.745587 1.00000 0
1.6 1.27571 −1.00000 −0.372560 0 −1.27571 2.62160 −3.02670 1.00000 0
1.7 2.04884 −1.00000 2.19776 0 −2.04884 −3.70442 0.405171 1.00000 0
1.8 2.09438 −1.00000 2.38641 0 −2.09438 −2.59420 0.809299 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 9075.2.a.dt 8
5.b even 2 1 9075.2.a.dw 8
11.b odd 2 1 9075.2.a.dv 8
11.d odd 10 2 825.2.n.n yes 16
55.d odd 2 1 9075.2.a.du 8
55.h odd 10 2 825.2.n.m ✓ 16
55.l even 20 4 825.2.bx.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.m ✓ 16 55.h odd 10 2
825.2.n.n yes 16 11.d odd 10 2
825.2.bx.j 32 55.l even 20 4
9075.2.a.dt 8 1.a even 1 1 trivial
9075.2.a.du 8 55.d odd 2 1
9075.2.a.dv 8 11.b odd 2 1
9075.2.a.dw 8 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 9T_{2}^{5} + 38T_{2}^{4} + 25T_{2}^{3} - 41T_{2}^{2} - 20T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{8} + 8T_{7}^{7} + 3T_{7}^{6} - 105T_{7}^{5} - 165T_{7}^{4} + 401T_{7}^{3} + 807T_{7}^{2} - 394T_{7} - 956 \) Copy content Toggle raw display
\( T_{13}^{8} + 7T_{13}^{7} - 69T_{13}^{6} - 541T_{13}^{5} + 908T_{13}^{4} + 11144T_{13}^{3} + 9019T_{13}^{2} - 44622T_{13} - 62716 \) Copy content Toggle raw display
\( T_{17}^{8} - T_{17}^{7} - 79T_{17}^{6} + 25T_{17}^{5} + 2019T_{17}^{4} + 598T_{17}^{3} - 17295T_{17}^{2} - 10532T_{17} + 31364 \) Copy content Toggle raw display
\( T_{19}^{8} - 6T_{19}^{7} - 55T_{19}^{6} + 503T_{19}^{5} - 809T_{19}^{4} - 2617T_{19}^{3} + 9969T_{19}^{2} - 10970T_{19} + 3980 \) Copy content Toggle raw display
\( T_{23}^{8} + 7T_{23}^{7} - 80T_{23}^{6} - 548T_{23}^{5} + 431T_{23}^{4} + 4440T_{23}^{3} - 1036T_{23}^{2} - 7409T_{23} + 3169 \) Copy content Toggle raw display
\( T_{37}^{8} + 14 T_{37}^{7} - 71 T_{37}^{6} - 1657 T_{37}^{5} - 3023 T_{37}^{4} + 34702 T_{37}^{3} + \cdots - 542461 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 11 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots - 956 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + \cdots - 62716 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + \cdots + 31364 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 3980 \) Copy content Toggle raw display
$23$ \( T^{8} + 7 T^{7} + \cdots + 3169 \) Copy content Toggle raw display
$29$ \( T^{8} - 28 T^{7} + \cdots - 49849 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 30689 \) Copy content Toggle raw display
$37$ \( T^{8} + 14 T^{7} + \cdots - 542461 \) Copy content Toggle raw display
$41$ \( T^{8} - 14 T^{7} + \cdots - 2253521 \) Copy content Toggle raw display
$43$ \( T^{8} + 11 T^{7} + \cdots + 5045 \) Copy content Toggle raw display
$47$ \( T^{8} + 5 T^{7} + \cdots + 4632625 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + \cdots - 16004 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + \cdots - 59695 \) Copy content Toggle raw display
$61$ \( T^{8} - 36 T^{7} + \cdots + 180080 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + \cdots + 45905 \) Copy content Toggle raw display
$71$ \( T^{8} + 17 T^{7} + \cdots + 123920 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots + 994769 \) Copy content Toggle raw display
$79$ \( T^{8} - 41 T^{7} + \cdots + 1546669 \) Copy content Toggle raw display
$83$ \( T^{8} - 6 T^{7} + \cdots + 498884 \) Copy content Toggle raw display
$89$ \( T^{8} - 21 T^{7} + \cdots - 60824801 \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 45758221 \) Copy content Toggle raw display
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