Properties

Label 2442.4.a.o
Level $2442$
Weight $4$
Character orbit 2442.a
Self dual yes
Analytic conductor $144.083$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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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 = [12,24,-36,48,15] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 987 x^{10} + 1990 x^{9} + 356253 x^{8} - 389456 x^{7} - 58119076 x^{6} + \cdots - 100856276127 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 11 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_1 + 1) q^{5} - 6 q^{6} + (\beta_{2} + 3) q^{7} + 8 q^{8} + 9 q^{9} + (2 \beta_1 + 2) q^{10} + 11 q^{11} - 12 q^{12} + (\beta_{10} + 5) q^{13} + (2 \beta_{2} + 6) q^{14}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{2} - 36 q^{3} + 48 q^{4} + 15 q^{5} - 72 q^{6} + 31 q^{7} + 96 q^{8} + 108 q^{9} + 30 q^{10} + 132 q^{11} - 144 q^{12} + 58 q^{13} + 62 q^{14} - 45 q^{15} + 192 q^{16} + 193 q^{17} + 216 q^{18}+ \cdots + 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 987 x^{10} + 1990 x^{9} + 356253 x^{8} - 389456 x^{7} - 58119076 x^{6} + \cdots - 100856276127 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 90\!\cdots\!85 \nu^{11} + \cdots - 22\!\cdots\!93 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 49\!\cdots\!74 \nu^{11} + \cdots + 33\!\cdots\!47 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!35 \nu^{11} + \cdots - 67\!\cdots\!07 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!06 \nu^{11} + \cdots - 49\!\cdots\!11 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34\!\cdots\!53 \nu^{11} + \cdots + 69\!\cdots\!19 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36\!\cdots\!68 \nu^{11} + \cdots + 29\!\cdots\!71 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55\!\cdots\!37 \nu^{11} + \cdots - 36\!\cdots\!48 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!99 \nu^{11} + \cdots + 49\!\cdots\!08 ) / 23\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!15 \nu^{11} + \cdots + 21\!\cdots\!94 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 93\!\cdots\!52 \nu^{11} + \cdots - 12\!\cdots\!63 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + 2\beta_{2} + \beta _1 + 166 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} + 22 \beta_{10} - 4 \beta_{9} - 9 \beta_{8} + 2 \beta_{7} - 15 \beta_{6} - 11 \beta_{5} + \cdots + 196 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 41 \beta_{11} - 173 \beta_{10} + 152 \beta_{9} + 426 \beta_{8} - 9 \beta_{7} - 365 \beta_{6} + \cdots + 44898 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 444 \beta_{11} + 9321 \beta_{10} - 1820 \beta_{9} - 4732 \beta_{8} + 1253 \beta_{7} - 7366 \beta_{6} + \cdots + 107379 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 23026 \beta_{11} - 92571 \beta_{10} + 90804 \beta_{9} + 169005 \beta_{8} + 1215 \beta_{7} + \cdots + 14301946 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4017 \beta_{11} + 3412424 \beta_{10} - 616156 \beta_{9} - 1992731 \beta_{8} + 630124 \beta_{7} + \cdots + 40648328 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10919421 \beta_{11} - 40082655 \beta_{10} + 42908956 \beta_{9} + 65901066 \beta_{8} + \cdots + 4953085339 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 63464475 \beta_{11} + 1230088586 \beta_{10} - 176601588 \beta_{9} - 787901618 \beta_{8} + \cdots + 14105530601 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4938179961 \beta_{11} - 16415182797 \beta_{10} + 18728381348 \beta_{9} + 25606930343 \beta_{8} + \cdots + 1794260324305 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 44651462692 \beta_{11} + 448041389353 \beta_{10} - 42421375052 \beta_{9} - 304288511333 \beta_{8} + \cdots + 4860849569159 \) Copy content Toggle raw display

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
−19.6270
−14.2381
−11.9335
−10.9766
−9.64027
0.706312
1.43724
5.90812
9.27840
15.1060
17.1221
19.8573
2.00000 −3.00000 4.00000 −18.6270 −6.00000 12.8007 8.00000 9.00000 −37.2541
1.2 2.00000 −3.00000 4.00000 −13.2381 −6.00000 −7.26950 8.00000 9.00000 −26.4761
1.3 2.00000 −3.00000 4.00000 −10.9335 −6.00000 −16.0653 8.00000 9.00000 −21.8671
1.4 2.00000 −3.00000 4.00000 −9.97656 −6.00000 12.9124 8.00000 9.00000 −19.9531
1.5 2.00000 −3.00000 4.00000 −8.64027 −6.00000 −18.6915 8.00000 9.00000 −17.2805
1.6 2.00000 −3.00000 4.00000 1.70631 −6.00000 16.8880 8.00000 9.00000 3.41262
1.7 2.00000 −3.00000 4.00000 2.43724 −6.00000 28.1775 8.00000 9.00000 4.87447
1.8 2.00000 −3.00000 4.00000 6.90812 −6.00000 −33.2414 8.00000 9.00000 13.8162
1.9 2.00000 −3.00000 4.00000 10.2784 −6.00000 18.3413 8.00000 9.00000 20.5568
1.10 2.00000 −3.00000 4.00000 16.1060 −6.00000 −21.1267 8.00000 9.00000 32.2120
1.11 2.00000 −3.00000 4.00000 18.1221 −6.00000 2.75551 8.00000 9.00000 36.2441
1.12 2.00000 −3.00000 4.00000 20.8573 −6.00000 35.5190 8.00000 9.00000 41.7147
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2442.4.a.o 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 15 T_{5}^{11} - 888 T_{5}^{10} + 11475 T_{5}^{9} + 294918 T_{5}^{8} - 3051182 T_{5}^{7} + \cdots - 417764713696 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2442))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{12} \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 417764713696 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 216453200304864 \) Copy content Toggle raw display
$11$ \( (T - 11)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 44\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 16\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 41\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T + 37)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 31\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 66\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 13\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 29\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 94\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 32\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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