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Properties

Label 1125.4.a.p
Level $1125$
Weight $4$
Character orbit 1125.a
Self dual yes
Analytic conductor $66.377$
Analytic rank $0$
Dimension $8$
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] = [1125,4,Mod(1,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1125 = 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,7,0,41,0,0,-19,99,0,0,-27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.3771487565\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 49x^{6} + 19x^{5} + 711x^{4} - 70x^{3} - 3215x^{2} + 4400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: no (minimal twist has level 375)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{7} + \beta_{5} + \beta_1 - 3) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots + 12) q^{8} + (3 \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 2) q^{11}+ \cdots + (28 \beta_{7} + 31 \beta_{6} + \cdots + 161) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} + 41 q^{4} - 19 q^{7} + 99 q^{8} - 27 q^{11} + 13 q^{13} - 119 q^{14} + 305 q^{16} + 223 q^{17} + 304 q^{19} - 31 q^{22} + 168 q^{23} - 221 q^{26} - 97 q^{28} - 168 q^{29} + 383 q^{31} + 774 q^{32}+ \cdots + 1378 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 49x^{6} + 19x^{5} + 711x^{4} - 70x^{3} - 3215x^{2} + 4400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 25\nu^{6} + 397\nu^{5} - 883\nu^{4} - 4819\nu^{3} + 9342\nu^{2} + 11947\nu - 21820 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 31\nu^{6} - 487\nu^{5} + 1109\nu^{4} + 5945\nu^{3} - 11870\nu^{2} - 14933\nu + 27860 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} - 37\nu^{6} - 569\nu^{5} + 1295\nu^{4} + 6879\nu^{3} - 13550\nu^{2} - 17215\nu + 31604 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{7} - 31\nu^{6} - 483\nu^{5} + 1089\nu^{4} + 5853\nu^{3} - 11454\nu^{2} - 14661\nu + 26772 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{7} - 42\nu^{6} - 659\nu^{5} + 1472\nu^{4} + 7989\nu^{3} - 15449\nu^{2} - 20010\nu + 36072 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{3} + 2\beta_{2} + 22\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 6\beta_{6} - 4\beta_{5} - \beta_{4} + \beta_{3} + 30\beta_{2} + 49\beta _1 + 250 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} + 54\beta_{6} - 43\beta_{5} - 7\beta_{4} + 28\beta_{3} + 92\beta_{2} + 579\beta _1 + 481 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} + 310\beta_{6} - 220\beta_{5} - 57\beta_{4} + 57\beta_{3} + 873\beta_{2} + 1933\beta _1 + 6418 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -253\beta_{7} + 2119\beta_{6} - 1580\beta_{5} - 369\beta_{4} + 759\beta_{3} + 3507\beta_{2} + 16688\beta _1 + 19730 \) 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
5.76438
3.91962
2.02388
1.56163
−1.81780
−1.81976
−4.28149
−4.35047
−4.76438 0 14.6994 0 0 12.5296 −31.9183 0 0
1.2 −2.91962 0 0.524200 0 0 −12.0754 21.8265 0 0
1.3 −1.02388 0 −6.95167 0 0 −27.1237 15.3087 0 0
1.4 −0.561635 0 −7.68457 0 0 35.7946 8.80899 0 0
1.5 2.81780 0 −0.0599776 0 0 −29.9198 −22.7114 0 0
1.6 2.81976 0 −0.0489573 0 0 10.5082 −22.6961 0 0
1.7 5.28149 0 19.8941 0 0 13.4711 62.8186 0 0
1.8 5.35047 0 20.6275 0 0 −22.1846 67.5631 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 1125.4.a.p 8
3.b odd 2 1 375.4.a.g ✓ 8
5.b even 2 1 1125.4.a.l 8
15.d odd 2 1 375.4.a.h yes 8
15.e even 4 2 375.4.b.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.4.a.g ✓ 8 3.b odd 2 1
375.4.a.h yes 8 15.d odd 2 1
375.4.b.d 16 15.e even 4 2
1125.4.a.l 8 5.b even 2 1
1125.4.a.p 8 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1125))\):

\( T_{2}^{8} - 7T_{2}^{7} - 28T_{2}^{6} + 240T_{2}^{5} + 106T_{2}^{4} - 2005T_{2}^{3} + 303T_{2}^{2} + 3994T_{2} + 1796 \) Copy content Toggle raw display
\( T_{11}^{8} + 27 T_{11}^{7} - 7113 T_{11}^{6} - 191540 T_{11}^{5} + 13815596 T_{11}^{4} + \cdots + 20623190976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{7} + \cdots + 1796 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13802270400 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 20623190976 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 864121395456 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 232303388570836 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 28410750225625 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 76\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 52\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 31\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 39\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
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