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Properties

Label 2442.4.a.r
Level $2442$
Weight $4$
Character orbit 2442.a
Self dual yes
Analytic conductor $144.083$
Analytic rank $0$
Dimension $13$
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] = [2442,4,Mod(1,2442)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2442, 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("2442.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2442 = 2 \cdot 3 \cdot 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2442.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,26,39,52,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(144.082664234\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 6 x^{12} - 1119 x^{11} + 4279 x^{10} + 456973 x^{9} - 976382 x^{8} - 83303882 x^{7} + \cdots + 1190588781472 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_1 + 2) q^{5} + 6 q^{6} + ( - \beta_{3} + 3) q^{7} + 8 q^{8} + 9 q^{9} + ( - 2 \beta_1 + 4) q^{10} - 11 q^{11} + 12 q^{12} + (\beta_{12} - \beta_{3} + 10) q^{13}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 26 q^{2} + 39 q^{3} + 52 q^{4} + 20 q^{5} + 78 q^{6} + 39 q^{7} + 104 q^{8} + 117 q^{9} + 40 q^{10} - 143 q^{11} + 156 q^{12} + 126 q^{13} + 78 q^{14} + 60 q^{15} + 208 q^{16} + 144 q^{17} + 234 q^{18}+ \cdots - 1287 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 6 x^{12} - 1119 x^{11} + 4279 x^{10} + 456973 x^{9} - 976382 x^{8} - 83303882 x^{7} + \cdots + 1190588781472 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 70\!\cdots\!67 \nu^{12} + \cdots + 10\!\cdots\!56 ) / 16\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\!\cdots\!85 \nu^{12} + \cdots + 11\!\cdots\!40 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!07 \nu^{12} + \cdots + 72\!\cdots\!64 ) / 82\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{12} + \cdots - 41\!\cdots\!36 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!94 \nu^{12} + \cdots + 16\!\cdots\!24 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!65 \nu^{12} + \cdots - 30\!\cdots\!04 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!79 \nu^{12} + \cdots + 30\!\cdots\!80 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!38 \nu^{12} + \cdots + 95\!\cdots\!76 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!49 \nu^{12} + \cdots - 21\!\cdots\!12 ) / 16\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46\!\cdots\!99 \nu^{12} + \cdots - 24\!\cdots\!56 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 48\!\cdots\!96 \nu^{12} + \cdots + 18\!\cdots\!24 ) / 33\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - \beta_{2} + 2\beta _1 + 175 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13 \beta_{12} + 14 \beta_{11} + \beta_{10} - 9 \beta_{9} + 4 \beta_{8} - 5 \beta_{7} - 17 \beta_{6} + \cdots + 442 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 521 \beta_{12} + 519 \beta_{11} + 494 \beta_{10} - 446 \beta_{9} + 394 \beta_{8} - 419 \beta_{7} + \cdots + 56372 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9232 \beta_{12} + 10684 \beta_{11} + 1720 \beta_{10} - 6167 \beta_{9} + 4551 \beta_{8} - 3897 \beta_{7} + \cdots + 353162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 240501 \beta_{12} + 255458 \beta_{11} + 220948 \beta_{10} - 204204 \beta_{9} + 182930 \beta_{8} + \cdots + 21998604 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5052940 \beta_{12} + 6077977 \beta_{11} + 1438379 \beta_{10} - 3385978 \beta_{9} + 3190799 \beta_{8} + \cdots + 216783210 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111800550 \beta_{12} + 128092247 \beta_{11} + 98424508 \beta_{10} - 96746197 \beta_{9} + \cdots + 9367010368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2567641805 \beta_{12} + 3188954961 \beta_{11} + 947600539 \beta_{10} - 1765956423 \beta_{9} + \cdots + 120460866624 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 53295402541 \beta_{12} + 65251975472 \beta_{11} + 44746247377 \beta_{10} - 46776011009 \beta_{9} + \cdots + 4188714644274 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1273280457041 \beta_{12} + 1630627656193 \beta_{11} + 559686771404 \beta_{10} - 907375093332 \beta_{9} + \cdots + 63949032885222 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25967438498422 \beta_{12} + 33511235000266 \beta_{11} + 20848254909424 \beta_{10} + \cdots + 19\!\cdots\!54 \) 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
22.6588
20.2455
12.4916
9.71948
8.57957
3.73627
−0.566027
−3.63640
−6.86225
−10.8674
−13.2697
−17.2466
−18.9829
2.00000 3.00000 4.00000 −20.6588 6.00000 −7.84056 8.00000 9.00000 −41.3177
1.2 2.00000 3.00000 4.00000 −18.2455 6.00000 32.0806 8.00000 9.00000 −36.4911
1.3 2.00000 3.00000 4.00000 −10.4916 6.00000 −2.61513 8.00000 9.00000 −20.9833
1.4 2.00000 3.00000 4.00000 −7.71948 6.00000 −4.84111 8.00000 9.00000 −15.4390
1.5 2.00000 3.00000 4.00000 −6.57957 6.00000 10.6215 8.00000 9.00000 −13.1591
1.6 2.00000 3.00000 4.00000 −1.73627 6.00000 −36.4082 8.00000 9.00000 −3.47254
1.7 2.00000 3.00000 4.00000 2.56603 6.00000 −19.8825 8.00000 9.00000 5.13205
1.8 2.00000 3.00000 4.00000 5.63640 6.00000 28.3482 8.00000 9.00000 11.2728
1.9 2.00000 3.00000 4.00000 8.86225 6.00000 10.5477 8.00000 9.00000 17.7245
1.10 2.00000 3.00000 4.00000 12.8674 6.00000 29.8264 8.00000 9.00000 25.7347
1.11 2.00000 3.00000 4.00000 15.2697 6.00000 12.8045 8.00000 9.00000 30.5395
1.12 2.00000 3.00000 4.00000 19.2466 6.00000 5.23980 8.00000 9.00000 38.4933
1.13 2.00000 3.00000 4.00000 20.9829 6.00000 −18.8811 8.00000 9.00000 41.9658
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)
\(37\) \( +1 \)

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 2442.4.a.r ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2442.4.a.r ✓ 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{13} - 20 T_{5}^{12} - 951 T_{5}^{11} + 19635 T_{5}^{10} + 297253 T_{5}^{9} - 6535936 T_{5}^{8} + \cdots - 3546965287456 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2442))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{13} \) Copy content Toggle raw display
$3$ \( (T - 3)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 3546965287456 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 276612142860160 \) Copy content Toggle raw display
$11$ \( (T + 11)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 35\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 12\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( (T + 37)^{13} \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 40\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 32\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 33\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 69\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 25\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 15\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 42\!\cdots\!60 \) Copy content Toggle raw display
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