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Properties

Label 2883.2.a.s
Level $2883$
Weight $2$
Character orbit 2883.a
Self dual yes
Analytic conductor $23.021$
Analytic rank $0$
Dimension $12$
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] = [2883,2,Mod(1,2883)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2883, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2883.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2883 = 3 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2883.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,5,-12,13,6] 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: \(23.0208709027\)
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: \( x^{12} - 5 x^{11} - 6 x^{10} + 57 x^{9} - 9 x^{8} - 229 x^{7} + 83 x^{6} + 436 x^{5} - 111 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{5} + \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{9} - \beta_{6}) q^{7} + (\beta_{6} + \beta_{5} + \beta_{2} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{8} + \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{2} - 12 q^{3} + 13 q^{4} + 6 q^{5} - 5 q^{6} + 6 q^{7} + 24 q^{8} + 12 q^{9} + 21 q^{10} - 2 q^{11} - 13 q^{12} - 3 q^{13} + 15 q^{14} - 6 q^{15} + 19 q^{16} + 8 q^{17} + 5 q^{18} + 15 q^{19}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 6 x^{10} + 57 x^{9} - 9 x^{8} - 229 x^{7} + 83 x^{6} + 436 x^{5} - 111 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} + 17 \nu^{10} + 4 \nu^{9} - 163 \nu^{8} + 157 \nu^{7} + 473 \nu^{6} - 607 \nu^{5} + \cdots - 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 7 \nu^{10} + 4 \nu^{9} + 65 \nu^{8} - 107 \nu^{7} - 179 \nu^{6} + 377 \nu^{5} + 210 \nu^{4} + \cdots + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 6 \nu^{10} + \nu^{9} + 53 \nu^{8} - 72 \nu^{7} - 128 \nu^{6} + 246 \nu^{5} + 109 \nu^{4} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} - \nu^{9} - 53 \nu^{8} + 72 \nu^{7} + 128 \nu^{6} - 246 \nu^{5} - 109 \nu^{4} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{11} - 33 \nu^{10} + 14 \nu^{9} + 297 \nu^{8} - 453 \nu^{7} - 757 \nu^{6} + 1515 \nu^{5} + \cdots + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{11} + 13 \nu^{10} - 5 \nu^{9} - 116 \nu^{8} + 173 \nu^{7} + 291 \nu^{6} - 573 \nu^{5} + \cdots - 32 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{11} + 65 \nu^{10} - 599 \nu^{8} + 693 \nu^{7} + 1613 \nu^{6} - 2435 \nu^{5} - 1750 \nu^{4} + \cdots - 152 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} + 33 \nu^{10} - 14 \nu^{9} - 297 \nu^{8} + 453 \nu^{7} + 761 \nu^{6} - 1523 \nu^{5} + \cdots - 88 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{11} + 104 \nu^{10} - 15 \nu^{9} - 949 \nu^{8} + 1220 \nu^{7} + 2496 \nu^{6} - 4216 \nu^{5} + \cdots - 232 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{2} + 8\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} + 8 \beta_{6} + 9 \beta_{5} + \beta_{4} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} + 11 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 4 \beta_{7} + 11 \beta_{6} + 13 \beta_{5} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{11} + 16 \beta_{10} - 16 \beta_{9} - 27 \beta_{8} + 18 \beta_{7} + 58 \beta_{6} + 71 \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{11} + 94 \beta_{10} - 39 \beta_{9} - 109 \beta_{8} + 61 \beta_{7} + 97 \beta_{6} + 128 \beta_{5} + \cdots + 419 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 146 \beta_{11} + 169 \beta_{10} - 182 \beta_{9} - 260 \beta_{8} + 212 \beta_{7} + 416 \beta_{6} + \cdots + 494 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 373 \beta_{11} + 746 \beta_{10} - 483 \beta_{9} - 894 \beta_{8} + 665 \beta_{7} + 792 \beta_{6} + \cdots + 2380 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1361 \beta_{11} + 1530 \beta_{10} - 1811 \beta_{9} - 2206 \beta_{8} + 2107 \beta_{7} + 2987 \beta_{6} + \cdots + 2918 \) 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
−2.35531
−1.35278
−1.14665
−1.12186
−0.766947
−0.135601
0.904643
1.17367
2.16596
2.27932
2.58026
2.77528
−2.35531 −1.00000 3.54749 1.29282 2.35531 3.92614 −3.64482 1.00000 −3.04499
1.2 −1.35278 −1.00000 −0.169991 −3.26657 1.35278 1.42631 2.93552 1.00000 4.41894
1.3 −1.14665 −1.00000 −0.685202 −0.861458 1.14665 −3.66740 3.07898 1.00000 0.987788
1.4 −1.12186 −1.00000 −0.741432 −1.73740 1.12186 −0.583356 3.07550 1.00000 1.94912
1.5 −0.766947 −1.00000 −1.41179 −1.16548 0.766947 −3.24795 2.61666 1.00000 0.893864
1.6 −0.135601 −1.00000 −1.98161 2.85131 0.135601 −2.89040 0.539910 1.00000 −0.386640
1.7 0.904643 −1.00000 −1.18162 −0.384739 −0.904643 3.36796 −2.87823 1.00000 −0.348052
1.8 1.17367 −1.00000 −0.622509 4.07994 −1.17367 2.16720 −3.07795 1.00000 4.78848
1.9 2.16596 −1.00000 2.69139 2.49070 −2.16596 1.25109 1.49753 1.00000 5.39476
1.10 2.27932 −1.00000 3.19531 0.797224 −2.27932 4.57095 2.72449 1.00000 1.81713
1.11 2.58026 −1.00000 4.65777 3.86423 −2.58026 −3.10198 6.85774 1.00000 9.97074
1.12 2.77528 −1.00000 5.70220 −1.96057 −2.77528 2.78143 10.2747 1.00000 −5.44114
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(31\) \( -1 \)

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 2883.2.a.s 12
3.b odd 2 1 8649.2.a.bl 12
31.b odd 2 1 2883.2.a.t 12
31.h odd 30 2 93.2.m.b ✓ 24
93.c even 2 1 8649.2.a.bk 12
93.p even 30 2 279.2.y.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.m.b ✓ 24 31.h odd 30 2
279.2.y.d 24 93.p even 30 2
2883.2.a.s 12 1.a even 1 1 trivial
2883.2.a.t 12 31.b odd 2 1
8649.2.a.bk 12 93.c even 2 1
8649.2.a.bl 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2883))\):

\( T_{2}^{12} - 5 T_{2}^{11} - 6 T_{2}^{10} + 57 T_{2}^{9} - 9 T_{2}^{8} - 229 T_{2}^{7} + 83 T_{2}^{6} + \cdots + 16 \) Copy content Toggle raw display
\( T_{11}^{12} + 2 T_{11}^{11} - 76 T_{11}^{10} - 107 T_{11}^{9} + 1986 T_{11}^{8} + 1765 T_{11}^{7} + \cdots - 23600 \) Copy content Toggle raw display
\( T_{13}^{12} + 3 T_{13}^{11} - 92 T_{13}^{10} - 255 T_{13}^{9} + 2994 T_{13}^{8} + 7551 T_{13}^{7} + \cdots - 152289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 496 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots - 40505 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots - 23600 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots - 152289 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 86256 \) Copy content Toggle raw display
$19$ \( T^{12} - 15 T^{11} + \cdots + 1795 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 9646576 \) Copy content Toggle raw display
$29$ \( T^{12} - 14 T^{11} + \cdots + 1974096 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 13 T^{11} + \cdots - 47281529 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + \cdots + 39554416 \) Copy content Toggle raw display
$43$ \( T^{12} - 19 T^{11} + \cdots - 11700155 \) Copy content Toggle raw display
$47$ \( T^{12} - 16 T^{11} + \cdots - 20809584 \) Copy content Toggle raw display
$53$ \( T^{12} - 20 T^{11} + \cdots + 2505520 \) Copy content Toggle raw display
$59$ \( T^{12} - 16 T^{11} + \cdots - 4615280 \) Copy content Toggle raw display
$61$ \( T^{12} + 35 T^{11} + \cdots - 63412205 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 618403131 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 1650238064 \) Copy content Toggle raw display
$73$ \( T^{12} + 27 T^{11} + \cdots - 4523579 \) Copy content Toggle raw display
$79$ \( T^{12} + 14 T^{11} + \cdots + 3866545 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 19674037904 \) Copy content Toggle raw display
$89$ \( T^{12} + 4 T^{11} + \cdots + 49526320 \) Copy content Toggle raw display
$97$ \( T^{12} - 12 T^{11} + \cdots + 30985501 \) Copy content Toggle raw display
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