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Properties

Label 1452.4.a.r
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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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] = [1452,4,Mod(1,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-18,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 74x^{4} - 18x^{3} + 1206x^{2} + 2088x - 99 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta_{4} - 2) q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + 9 q^{9} + ( - \beta_{3} - 5 \beta_1) q^{13} + (3 \beta_{4} + 6) q^{15} + (3 \beta_{3} - 2 \beta_{2} + 11 \beta_1) q^{17} + (2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{19}+ \cdots + ( - 28 \beta_{5} - 707) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{3} - 12 q^{5} + 54 q^{9} + 36 q^{15} - 492 q^{23} + 534 q^{25} - 162 q^{27} - 270 q^{31} + 582 q^{37} - 108 q^{45} + 756 q^{47} + 780 q^{49} - 72 q^{53} + 564 q^{59} - 1590 q^{67} + 1476 q^{69}+ \cdots - 4242 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 74x^{4} - 18x^{3} + 1206x^{2} + 2088x - 99 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 128\nu^{5} - 745\nu^{4} - 7540\nu^{3} + 27720\nu^{2} + 94785\nu - 71991 ) / 39003 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 989\nu^{5} - 6061\nu^{4} - 44851\nu^{3} + 216618\nu^{2} + 193938\nu - 1189737 ) / 39003 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 365\nu^{5} - 804\nu^{4} - 27595\nu^{3} + 16478\nu^{2} + 419289\nu + 199674 ) / 13001 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1106\nu^{5} - 4609\nu^{4} - 67588\nu^{3} + 107883\nu^{2} + 812298\nu + 643722 ) / 39003 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1618\nu^{5} - 7589\nu^{4} - 97748\nu^{3} + 218763\nu^{2} + 1347450\nu + 303754 ) / 13001 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 3\beta_{4} - 12\beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 15\beta_{4} + 6\beta_{3} + 6\beta_{2} - 6\beta _1 + 304 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 43\beta_{5} - 141\beta_{4} + 9\beta_{3} + 45\beta_{2} - 837\beta _1 + 1012 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 133\beta_{5} - 1023\beta_{4} + 444\beta_{3} + 492\beta_{2} - 3804\beta _1 + 14944 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2350\beta_{5} - 8790\beta_{4} + 1815\beta_{3} + 4215\beta_{2} - 57603\beta _1 + 84544 ) / 12 \) 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
0.0461829
−3.41792
−5.89143
−2.42733
8.57730
5.11320
0 −3.00000 0 −19.4720 0 −20.3144 0 9.00000 0
1.2 0 −3.00000 0 −19.4720 0 20.3144 0 9.00000 0
1.3 0 −3.00000 0 −2.54018 0 −26.1494 0 9.00000 0
1.4 0 −3.00000 0 −2.54018 0 26.1494 0 9.00000 0
1.5 0 −3.00000 0 16.0122 0 −17.9593 0 9.00000 0
1.6 0 −3.00000 0 16.0122 0 17.9593 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(11\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.4.a.r ✓ 6
11.b odd 2 1 inner 1452.4.a.r ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.r ✓ 6 1.a even 1 1 trivial
1452.4.a.r ✓ 6 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{3} + 6T_{5}^{2} - 303T_{5} - 792 \) Copy content Toggle raw display
\( T_{7}^{6} - 1419T_{7}^{4} + 635832T_{7}^{2} - 91014192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + 6 T^{2} + \cdots - 792)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 1419 T^{4} + \cdots - 91014192 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 1566 T^{4} + \cdots - 3345408 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 137182636800 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 142028391168 \) Copy content Toggle raw display
$23$ \( (T^{3} + 246 T^{2} + \cdots - 1937592)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 8567590214700 \) Copy content Toggle raw display
$31$ \( (T^{3} + 135 T^{2} + \cdots - 10328560)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 291 T^{2} + \cdots + 6033887)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 515723440485132 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 23672791339200 \) Copy content Toggle raw display
$47$ \( (T^{3} - 378 T^{2} + \cdots + 17449344)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 36 T^{2} + \cdots - 2972862)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 282 T^{2} + \cdots - 7260984)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 673929065115648 \) Copy content Toggle raw display
$67$ \( (T^{3} + 795 T^{2} + \cdots - 103881100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 492 T^{2} + \cdots + 26181936)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 38\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 585991094796288 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1662 T^{2} + \cdots - 810794916)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2121 T^{2} + \cdots - 1565623157)^{2} \) Copy content Toggle raw display
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