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Properties

Label 2142.2.p.h
Level $2142$
Weight $2$
Character orbit 2142.p
Analytic conductor $17.104$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2142,2,Mod(1135,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) 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("2142.1135"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2142.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-16,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.1039561130\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 36 x^{14} - 108 x^{13} + 244 x^{12} - 424 x^{11} + 674 x^{10} - 1804 x^{9} + \cdots + 388 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - q^{4} - \beta_{5} q^{5} - \beta_{9} q^{7} - \beta_{12} q^{8} + \beta_{10} q^{10} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots - 1) q^{11} + ( - \beta_{14} + \beta_{13} - \beta_{7} + \cdots - 1) q^{13}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 4 q^{11} - 24 q^{13} + 16 q^{16} + 8 q^{17} - 4 q^{22} - 4 q^{29} + 24 q^{31} + 8 q^{34} + 8 q^{35} - 12 q^{37} + 8 q^{38} + 4 q^{44} - 32 q^{47} - 8 q^{50} + 24 q^{52} - 16 q^{55} + 4 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 36 x^{14} - 108 x^{13} + 244 x^{12} - 424 x^{11} + 674 x^{10} - 1804 x^{9} + \cdots + 388 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 83\!\cdots\!88 \nu^{15} + \cdots - 76\!\cdots\!48 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49\!\cdots\!02 \nu^{15} + \cdots - 33\!\cdots\!74 ) / 72\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!40 \nu^{15} + \cdots + 10\!\cdots\!88 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!00 \nu^{15} + \cdots - 11\!\cdots\!46 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59\!\cdots\!96 \nu^{15} + \cdots - 33\!\cdots\!10 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 69\!\cdots\!81 \nu^{15} + \cdots - 31\!\cdots\!84 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 72\!\cdots\!75 \nu^{15} + \cdots - 44\!\cdots\!08 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!33 \nu^{15} + \cdots + 42\!\cdots\!50 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 90\!\cdots\!97 \nu^{15} + \cdots - 42\!\cdots\!74 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 91\!\cdots\!28 \nu^{15} + \cdots + 41\!\cdots\!70 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!66 \nu^{15} + \cdots + 52\!\cdots\!06 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13722817214568 \nu^{15} + 98291343706372 \nu^{14} - 411624462299224 \nu^{13} + \cdots + 65\!\cdots\!30 ) / 160550220964286 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!05 \nu^{15} + \cdots + 17\!\cdots\!22 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43\!\cdots\!39 \nu^{15} + \cdots + 17\!\cdots\!22 ) / 14\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 55\!\cdots\!77 \nu^{15} + \cdots + 24\!\cdots\!86 ) / 14\!\cdots\!06 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{10} + 5 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} + 4 \beta_{11} + 5 \beta_{10} + 6 \beta_{8} - 4 \beta_{7} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 4 \beta_{14} + 5 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 7 \beta_{10} - 7 \beta_{8} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{14} + 54 \beta_{12} - 26 \beta_{11} + 45 \beta_{10} + 66 \beta_{9} - 26 \beta_{8} - 11 \beta_{6} + \cdots - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 26 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} + 50 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 160 \beta_{15} - 57 \beta_{14} + 146 \beta_{13} + 30 \beta_{12} + 48 \beta_{11} - 71 \beta_{10} + \cdots + 1005 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 148 \beta_{15} + 112 \beta_{14} - 208 \beta_{13} - 244 \beta_{12} + 66 \beta_{11} - 430 \beta_{10} + \cdots + 608 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1716 \beta_{15} + 1829 \beta_{14} - 2690 \beta_{13} - 1356 \beta_{12} - 182 \beta_{11} - 3687 \beta_{10} + \cdots - 3099 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1188 \beta_{15} + 799 \beta_{14} - 1631 \beta_{13} - 3720 \beta_{12} + 1482 \beta_{11} - 905 \beta_{10} + \cdots - 6703 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3138 \beta_{15} - 11405 \beta_{14} + 12574 \beta_{13} - 8552 \beta_{12} + 6024 \beta_{11} + \cdots - 29839 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6874 \beta_{15} - 16085 \beta_{14} + 20855 \beta_{13} + 24116 \beta_{12} - 9730 \beta_{11} + \cdots - 15132 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 46328 \beta_{15} - 11147 \beta_{14} + 42620 \beta_{13} + 204136 \beta_{12} - 84310 \beta_{11} + \cdots + 47047 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 94734 \beta_{15} + 23860 \beta_{14} + 32452 \beta_{13} + 143402 \beta_{12} - 33294 \beta_{11} + \cdots + 445649 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 110634 \beta_{15} + 168967 \beta_{14} - 143964 \beta_{13} - 472704 \beta_{12} + 312272 \beta_{11} + \cdots + 2923859 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2142\mathbb{Z}\right)^\times\).

\(n\) \(1261\) \(1667\) \(1837\)
\(\chi(n)\) \(\beta_{12}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1
−1.93087 − 1.21401i
1.28212 − 0.681097i
0.343636 + 1.58460i
1.49123 + 0.203473i
0.851331 − 0.0615806i
0.174102 − 0.342098i
−0.0885438 + 2.62798i
1.87700 − 2.11727i
−1.93087 + 1.21401i
1.28212 + 0.681097i
0.343636 − 1.58460i
1.49123 − 0.203473i
0.851331 + 0.0615806i
0.174102 + 0.342098i
−0.0885438 − 2.62798i
1.87700 + 2.11727i
1.00000i 0 −1.00000 −2.55711 − 2.55711i 0 −0.707107 + 0.707107i − 1.00000i 0 2.55711 − 2.55711i
1135.2 1.00000i 0 −1.00000 −1.63093 − 1.63093i 0 0.707107 − 0.707107i − 1.00000i 0 1.63093 − 1.63093i
1135.3 1.00000i 0 −1.00000 −1.26230 − 1.26230i 0 0.707107 − 0.707107i − 1.00000i 0 1.26230 − 1.26230i
1135.4 1.00000i 0 −1.00000 −0.480388 − 0.480388i 0 −0.707107 + 0.707107i − 1.00000i 0 0.480388 − 0.480388i
1135.5 1.00000i 0 −1.00000 0.606153 + 0.606153i 0 −0.707107 + 0.707107i − 1.00000i 0 −0.606153 + 0.606153i
1135.6 1.00000i 0 −1.00000 1.01713 + 1.01713i 0 −0.707107 + 0.707107i − 1.00000i 0 −1.01713 + 1.01713i
1135.7 1.00000i 0 −1.00000 1.76769 + 1.76769i 0 0.707107 − 0.707107i − 1.00000i 0 −1.76769 + 1.76769i
1135.8 1.00000i 0 −1.00000 2.53976 + 2.53976i 0 0.707107 − 0.707107i − 1.00000i 0 −2.53976 + 2.53976i
1891.1 − 1.00000i 0 −1.00000 −2.55711 + 2.55711i 0 −0.707107 − 0.707107i 1.00000i 0 2.55711 + 2.55711i
1891.2 − 1.00000i 0 −1.00000 −1.63093 + 1.63093i 0 0.707107 + 0.707107i 1.00000i 0 1.63093 + 1.63093i
1891.3 − 1.00000i 0 −1.00000 −1.26230 + 1.26230i 0 0.707107 + 0.707107i 1.00000i 0 1.26230 + 1.26230i
1891.4 − 1.00000i 0 −1.00000 −0.480388 + 0.480388i 0 −0.707107 − 0.707107i 1.00000i 0 0.480388 + 0.480388i
1891.5 − 1.00000i 0 −1.00000 0.606153 − 0.606153i 0 −0.707107 − 0.707107i 1.00000i 0 −0.606153 − 0.606153i
1891.6 − 1.00000i 0 −1.00000 1.01713 − 1.01713i 0 −0.707107 − 0.707107i 1.00000i 0 −1.01713 − 1.01713i
1891.7 − 1.00000i 0 −1.00000 1.76769 − 1.76769i 0 0.707107 + 0.707107i 1.00000i 0 −1.76769 − 1.76769i
1891.8 − 1.00000i 0 −1.00000 2.53976 − 2.53976i 0 0.707107 + 0.707107i 1.00000i 0 −2.53976 − 2.53976i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1135.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2142.2.p.h ✓ 16
3.b odd 2 1 2142.2.p.i yes 16
17.c even 4 1 inner 2142.2.p.h ✓ 16
51.f odd 4 1 2142.2.p.i yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2142.2.p.h ✓ 16 1.a even 1 1 trivial
2142.2.p.h ✓ 16 17.c even 4 1 inner
2142.2.p.i yes 16 3.b odd 2 1
2142.2.p.i yes 16 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 210 T_{5}^{12} + 32 T_{5}^{9} + 7201 T_{5}^{8} + 704 T_{5}^{7} - 4384 T_{5}^{5} + \cdots + 12544 \) acting on \(S_{2}^{\mathrm{new}}(2142, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 210 T^{12} + \cdots + 12544 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + \cdots + 38416 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + 24 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + 156 T^{14} + \cdots + 135424 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 8728043776 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 48809181184 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 600838144 \) Copy content Toggle raw display
$37$ \( T^{16} + 12 T^{15} + \cdots + 35344 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 205520896 \) Copy content Toggle raw display
$43$ \( T^{16} + 364 T^{14} + \cdots + 2027776 \) Copy content Toggle raw display
$47$ \( (T^{8} + 16 T^{7} + \cdots - 82944)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 140283207936 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 73879588864 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 17163096064 \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + \cdots + 129472)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 11053337518336 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1765114187776 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 128129634304 \) Copy content Toggle raw display
$83$ \( T^{16} + 488 T^{14} + \cdots + 446224 \) Copy content Toggle raw display
$89$ \( (T^{8} - 278 T^{6} + \cdots + 2937284)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
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