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Properties

Label 1547.2.a.i
Level $1547$
Weight $2$
Character orbit 1547.a
Self dual yes
Analytic conductor $12.353$
Analytic rank $1$
Dimension $14$
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] = [1547,2,Mod(1,1547)] 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("1547.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1547, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1547 = 7 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1547.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.3528571927\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 20 x^{12} + 19 x^{11} + 151 x^{10} - 133 x^{9} - 536 x^{8} + 404 x^{7} + 924 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - 1) q^{5} + ( - \beta_{4} - \beta_1 - 1) q^{6} - q^{7} + (\beta_{13} + \beta_{12} + \beta_{8} + \cdots + 1) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{10} + 1) q^{9}+ \cdots + ( - 2 \beta_{12} - \beta_{11} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} - 4 q^{3} + 13 q^{4} - 9 q^{5} - 10 q^{6} - 14 q^{7} + 18 q^{9} - 9 q^{10} + 3 q^{11} - 16 q^{12} - 14 q^{13} - q^{14} - q^{15} + 11 q^{16} - 14 q^{17} + 5 q^{18} - 17 q^{19} - 24 q^{20}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 20 x^{12} + 19 x^{11} + 151 x^{10} - 133 x^{9} - 536 x^{8} + 404 x^{7} + 924 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{12} - \nu^{11} + 23 \nu^{10} + 12 \nu^{9} - 190 \nu^{8} - 39 \nu^{7} + 684 \nu^{6} + 35 \nu^{5} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} - \nu^{12} - 21 \nu^{11} + 22 \nu^{10} + 162 \nu^{9} - 173 \nu^{8} - 558 \nu^{7} + 573 \nu^{6} + \cdots + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{13} - \nu^{12} - 19 \nu^{11} + 18 \nu^{10} + 134 \nu^{9} - 119 \nu^{8} - 430 \nu^{7} + 339 \nu^{6} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{13} + 3 \nu^{12} + 19 \nu^{11} - 56 \nu^{10} - 138 \nu^{9} + 389 \nu^{8} + 488 \nu^{7} + \cdots - 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{13} + 3 \nu^{12} + 39 \nu^{11} - 59 \nu^{10} - 280 \nu^{9} + 428 \nu^{8} + 899 \nu^{7} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{13} - 5 \nu^{12} - 37 \nu^{11} + 97 \nu^{10} + 244 \nu^{9} - 692 \nu^{8} - 665 \nu^{7} + 2170 \nu^{6} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{13} + 3 \nu^{12} + 59 \nu^{11} - 58 \nu^{10} - 430 \nu^{9} + 411 \nu^{8} + 1418 \nu^{7} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2 \nu^{13} - 7 \nu^{12} - 35 \nu^{11} + 131 \nu^{10} + 216 \nu^{9} - 904 \nu^{8} - 535 \nu^{7} + \cdots + 359 \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{13} + 5 \nu^{12} + 97 \nu^{11} - 94 \nu^{10} - 694 \nu^{9} + 645 \nu^{8} + 2222 \nu^{7} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{13} - 3 \nu^{12} - 57 \nu^{11} + 54 \nu^{10} + 400 \nu^{9} - 355 \nu^{8} - 1262 \nu^{7} + \cdots + 8 ) / 2 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{13} + 9 \beta_{12} - \beta_{10} + 9 \beta_{8} - \beta_{6} + 8 \beta_{5} + 9 \beta_{4} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{13} + 12 \beta_{12} - \beta_{10} + 11 \beta_{8} - 2 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68 \beta_{13} + 69 \beta_{12} + \beta_{11} - 14 \beta_{10} - \beta_{9} + 68 \beta_{8} + \beta_{7} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 109 \beta_{13} + 110 \beta_{12} - 16 \beta_{10} + \beta_{9} + 98 \beta_{8} - 27 \beta_{6} + 96 \beta_{5} + \cdots + 607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 492 \beta_{13} + 508 \beta_{12} + 14 \beta_{11} - 138 \beta_{10} - 13 \beta_{9} + 493 \beta_{8} + \cdots + 638 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 895 \beta_{13} + 914 \beta_{12} + \beta_{11} - 174 \beta_{10} + 15 \beta_{9} + 805 \beta_{8} + \cdots + 3961 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3515 \beta_{13} + 3686 \beta_{12} + 134 \beta_{11} - 1178 \beta_{10} - 115 \beta_{9} + 3529 \beta_{8} + \cdots + 5128 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7001 \beta_{13} + 7230 \beta_{12} + 18 \beta_{11} - 1613 \beta_{10} + 153 \beta_{9} + 6335 \beta_{8} + \cdots + 26429 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25031 \beta_{13} + 26572 \beta_{12} + 1100 \beta_{11} - 9334 \beta_{10} - 871 \beta_{9} + 25147 \beta_{8} + \cdots + 40189 \) 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
−2.44528
−2.33271
−2.31639
−1.27615
−0.650788
−0.551317
−0.308736
−0.0757200
1.17650
1.21390
1.64936
2.03743
2.19455
2.68534
−2.44528 −2.31796 3.97939 1.67233 5.66806 −1.00000 −4.84016 2.37294 −4.08931
1.2 −2.33271 −0.539163 3.44151 −3.18493 1.25771 −1.00000 −3.36263 −2.70930 7.42951
1.3 −2.31639 2.85462 3.36565 −0.549273 −6.61240 −1.00000 −3.16337 5.14883 1.27233
1.4 −1.27615 1.43058 −0.371430 0.109464 −1.82564 −1.00000 3.02631 −0.953436 −0.139693
1.5 −0.650788 −1.79830 −1.57647 0.841445 1.17031 −1.00000 2.32753 0.233881 −0.547603
1.6 −0.551317 1.13506 −1.69605 3.14704 −0.625779 −1.00000 2.03769 −1.71163 −1.73501
1.7 −0.308736 −2.95438 −1.90468 −3.87433 0.912122 −1.00000 1.20552 5.72834 1.19614
1.8 −0.0757200 2.32101 −1.99427 −2.99129 −0.175747 −1.00000 0.302446 2.38707 0.226501
1.9 1.17650 −2.07753 −0.615851 2.62922 −2.44422 −1.00000 −3.07755 1.31615 3.09328
1.10 1.21390 0.663918 −0.526452 1.56682 0.805928 −1.00000 −3.06685 −2.55921 1.90196
1.11 1.64936 2.46872 0.720402 −3.73461 4.07182 −1.00000 −2.11052 3.09457 −6.15973
1.12 2.03743 −0.0623213 2.15112 −0.431630 −0.126975 −1.00000 0.307904 −2.99612 −0.879416
1.13 2.19455 −3.43361 2.81607 −1.44694 −7.53523 −1.00000 1.79091 8.78965 −3.17540
1.14 2.68534 −1.69064 5.21106 −2.75330 −4.53995 −1.00000 8.62278 −0.141726 −7.39356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(13\) \( +1 \)
\(17\) \( +1 \)

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 1547.2.a.i ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1547.2.a.i ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1547))\):

\( T_{2}^{14} - T_{2}^{13} - 20 T_{2}^{12} + 19 T_{2}^{11} + 151 T_{2}^{10} - 133 T_{2}^{9} - 536 T_{2}^{8} + \cdots + 4 \) Copy content Toggle raw display
\( T_{3}^{14} + 4 T_{3}^{13} - 22 T_{3}^{12} - 95 T_{3}^{11} + 173 T_{3}^{10} + 850 T_{3}^{9} - 570 T_{3}^{8} + \cdots + 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - T^{13} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots + 88 \) Copy content Toggle raw display
$5$ \( T^{14} + 9 T^{13} + \cdots + 260 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 3 T^{13} + \cdots - 1540096 \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( (T + 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + 17 T^{13} + \cdots - 2964224 \) Copy content Toggle raw display
$23$ \( T^{14} + 6 T^{13} + \cdots - 3297344 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 297991936 \) Copy content Toggle raw display
$31$ \( T^{14} + 19 T^{13} + \cdots + 25640768 \) Copy content Toggle raw display
$37$ \( T^{14} + 45 T^{13} + \cdots - 2877440 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 77145674072 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 271301888 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 15773571328 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 260801548288 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 48879517696 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 8232577312 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 29928253952 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 87126278144 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 1393407583700 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1285842368 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 11746963456 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 190275139840 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 121699866292 \) Copy content Toggle raw display
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