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Properties

Label 2442.4.a.j
Level $2442$
Weight $4$
Character orbit 2442.a
Self dual yes
Analytic conductor $144.083$
Analytic rank $0$
Dimension $11$
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 = [11,-22,-33,44,5] 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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 780 x^{9} + 2999 x^{8} + 213314 x^{7} - 597996 x^{6} - 23648188 x^{5} + \cdots + 1127485056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 37 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + \beta_1 q^{5} + 6 q^{6} + ( - \beta_{7} + 2) q^{7} - 8 q^{8} + 9 q^{9} - 2 \beta_1 q^{10} + 11 q^{11} - 12 q^{12} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 22 q^{2} - 33 q^{3} + 44 q^{4} + 5 q^{5} + 66 q^{6} + 17 q^{7} - 88 q^{8} + 99 q^{9} - 10 q^{10} + 121 q^{11} - 132 q^{12} - 10 q^{13} - 34 q^{14} - 15 q^{15} + 176 q^{16} + 23 q^{17} - 198 q^{18}+ \cdots + 1089 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 780 x^{9} + 2999 x^{8} + 213314 x^{7} - 597996 x^{6} - 23648188 x^{5} + \cdots + 1127485056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!59 \nu^{10} + \cdots - 98\!\cdots\!16 ) / 35\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 46\!\cdots\!61 \nu^{10} + \cdots + 53\!\cdots\!36 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73\!\cdots\!19 \nu^{10} + \cdots - 24\!\cdots\!44 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!85 \nu^{10} + \cdots - 32\!\cdots\!20 ) / 64\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!33 \nu^{10} + \cdots + 51\!\cdots\!48 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!81 \nu^{10} + \cdots - 21\!\cdots\!56 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 79\!\cdots\!17 \nu^{10} + \cdots - 27\!\cdots\!88 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!43 \nu^{10} + \cdots - 42\!\cdots\!60 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!19 \nu^{10} + \cdots + 35\!\cdots\!96 ) / 89\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} + 3\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 143 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 20 \beta_{10} + 6 \beta_{9} + 19 \beta_{8} - 16 \beta_{7} - \beta_{6} - 7 \beta_{5} + 16 \beta_{4} + \cdots + 158 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 217 \beta_{10} - 156 \beta_{9} + 276 \beta_{8} - 207 \beta_{7} + 21 \beta_{6} + 948 \beta_{5} + \cdots + 34346 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7768 \beta_{10} + 1955 \beta_{9} + 7259 \beta_{8} - 7806 \beta_{7} - 1899 \beta_{6} - 1290 \beta_{5} + \cdots + 91364 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 101604 \beta_{10} - 6417 \beta_{9} + 73394 \beta_{8} - 76158 \beta_{7} + 698 \beta_{6} + 281711 \beta_{5} + \cdots + 9175808 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2566329 \beta_{10} + 734311 \beta_{9} + 2291548 \beta_{8} - 2676105 \beta_{7} - 1008483 \beta_{6} + \cdots + 42750162 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 37555687 \beta_{10} + 8752454 \beta_{9} + 19070199 \beta_{8} - 22444737 \beta_{7} - 3579488 \beta_{6} + \cdots + 2600678418 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 823792191 \beta_{10} + 298566683 \beta_{9} + 678737923 \beta_{8} - 826221975 \beta_{7} + \cdots + 17622814106 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12989781716 \beta_{10} + 5614001012 \beta_{9} + 4976489695 \beta_{8} - 6277428528 \beta_{7} + \cdots + 768826362276 \) 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
−16.4416
−14.7707
−14.0813
−4.95865
−4.00198
0.198573
2.33414
10.5399
11.0856
16.4799
18.6162
−2.00000 −3.00000 4.00000 −16.4416 6.00000 −15.8556 −8.00000 9.00000 32.8831
1.2 −2.00000 −3.00000 4.00000 −14.7707 6.00000 −4.87313 −8.00000 9.00000 29.5414
1.3 −2.00000 −3.00000 4.00000 −14.0813 6.00000 1.83327 −8.00000 9.00000 28.1626
1.4 −2.00000 −3.00000 4.00000 −4.95865 6.00000 −22.8460 −8.00000 9.00000 9.91730
1.5 −2.00000 −3.00000 4.00000 −4.00198 6.00000 31.8224 −8.00000 9.00000 8.00396
1.6 −2.00000 −3.00000 4.00000 0.198573 6.00000 13.1588 −8.00000 9.00000 −0.397147
1.7 −2.00000 −3.00000 4.00000 2.33414 6.00000 31.5315 −8.00000 9.00000 −4.66828
1.8 −2.00000 −3.00000 4.00000 10.5399 6.00000 −9.85561 −8.00000 9.00000 −21.0797
1.9 −2.00000 −3.00000 4.00000 11.0856 6.00000 −24.8427 −8.00000 9.00000 −22.1712
1.10 −2.00000 −3.00000 4.00000 16.4799 6.00000 15.8441 −8.00000 9.00000 −32.9597
1.11 −2.00000 −3.00000 4.00000 18.6162 6.00000 1.08307 −8.00000 9.00000 −37.2323
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.j ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2442.4.a.j ✓ 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{11} - 5 T_{5}^{10} - 780 T_{5}^{9} + 2999 T_{5}^{8} + 213314 T_{5}^{7} - 597996 T_{5}^{6} + \cdots + 1127485056 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2442))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{11} \) Copy content Toggle raw display
$3$ \( (T + 3)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots + 1127485056 \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots + 179525046272 \) Copy content Toggle raw display
$11$ \( (T - 11)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 39\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots - 82\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 72\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( (T - 37)^{11} \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 39\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 75\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 48\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 34\!\cdots\!80 \) Copy content Toggle raw display
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