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Properties

Label 2057.4.a.h
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $0$
Dimension $10$
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] = [2057,4,Mod(1,2057)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2057 = 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2057.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-4,11,40,47] 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: \(121.366928882\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 52 x^{8} + 180 x^{7} + 933 x^{6} - 2524 x^{5} - 6654 x^{4} + 11196 x^{3} + 16044 x^{2} + \cdots - 88 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 187)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + (\beta_{8} + \beta_{6} - \beta_{5} + 5) q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 10) q^{6} + (\beta_{9} - \beta_{8} - \beta_{6} + \cdots + 2) q^{7}+ \cdots + ( - 48 \beta_{9} + 34 \beta_{8} + \cdots - 332) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 11 q^{3} + 40 q^{4} + 47 q^{5} - 105 q^{6} + 17 q^{7} - 84 q^{8} + 107 q^{9} + 7 q^{10} + 101 q^{12} - 15 q^{13} + 45 q^{14} + 224 q^{15} + 140 q^{16} + 170 q^{17} - 81 q^{18} - q^{19}+ \cdots - 3281 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 52 x^{8} + 180 x^{7} + 933 x^{6} - 2524 x^{5} - 6654 x^{4} + 11196 x^{3} + 16044 x^{2} + \cdots - 88 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83 \nu^{9} + 2717 \nu^{8} - 19573 \nu^{7} - 113171 \nu^{6} + 713942 \nu^{5} + 1274510 \nu^{4} + \cdots - 6454184 ) / 574464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 265 \nu^{9} - 3113 \nu^{8} - 10799 \nu^{7} + 138407 \nu^{6} + 164578 \nu^{5} - 1879382 \nu^{4} + \cdots - 1179640 ) / 574464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 349 \nu^{9} + 1661 \nu^{8} + 15035 \nu^{7} - 73619 \nu^{6} - 187210 \nu^{5} + 1045454 \nu^{4} + \cdots + 3970648 ) / 574464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 491 \nu^{9} - 1771 \nu^{8} - 27325 \nu^{7} + 86341 \nu^{6} + 505862 \nu^{5} - 1278754 \nu^{4} + \cdots - 1678952 ) / 287232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 109 \nu^{9} + 440 \nu^{8} + 5657 \nu^{7} - 19982 \nu^{6} - 101710 \nu^{5} + 281216 \nu^{4} + \cdots + 536272 ) / 35904 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2305 \nu^{9} + 7601 \nu^{8} + 121367 \nu^{7} - 328127 \nu^{6} - 2176018 \nu^{5} + 4269446 \nu^{4} + \cdots + 3585208 ) / 574464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 973 \nu^{9} + 4125 \nu^{8} + 49067 \nu^{7} - 180051 \nu^{6} - 842602 \nu^{5} + 2398094 \nu^{4} + \cdots + 3677784 ) / 191488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} + 20\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 3\beta_{8} - 2\beta_{7} + 5\beta_{5} + \beta_{4} - 2\beta_{3} + 25\beta_{2} + 32\beta _1 + 236 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{9} - 33 \beta_{8} - 14 \beta_{7} - 8 \beta_{6} - 21 \beta_{5} + 31 \beta_{4} - 2 \beta_{3} + \cdots + 312 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 135 \beta_{9} - 107 \beta_{8} - 106 \beta_{7} + 4 \beta_{6} + 133 \beta_{5} + 41 \beta_{4} - 70 \beta_{3} + \cdots + 5224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1137 \beta_{9} - 933 \beta_{8} - 678 \beta_{7} - 376 \beta_{6} - 457 \beta_{5} + 779 \beta_{4} + \cdots + 10128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4683 \beta_{9} - 3319 \beta_{8} - 3946 \beta_{7} - 44 \beta_{6} + 2753 \beta_{5} + 1165 \beta_{4} + \cdots + 122472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 33201 \beta_{9} - 25973 \beta_{8} - 24110 \beta_{7} - 12960 \beta_{6} - 11313 \beta_{5} + 18091 \beta_{4} + \cdots + 306944 \) 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
5.25499
4.95094
3.74503
2.53711
0.417002
−0.0109263
−1.58216
−2.62662
−4.10090
−4.58446
−5.25499 6.75156 19.6149 4.41497 −35.4794 31.2925 −61.0362 18.5836 −23.2006
1.2 −4.95094 0.0975223 16.5118 7.22801 −0.482827 −10.3683 −42.1416 −26.9905 −35.7855
1.3 −3.74503 9.00843 6.02526 −6.24362 −33.7368 −21.2202 7.39547 54.1518 23.3826
1.4 −2.53711 5.15451 −1.56306 22.0279 −13.0776 −20.1356 24.2626 −0.431059 −55.8872
1.5 −0.417002 −5.36514 −7.82611 −14.2046 2.23727 −7.39855 6.59952 1.78467 5.92335
1.6 0.0109263 −5.88845 −7.99988 −1.15833 −0.0643390 32.9609 −0.174820 7.67385 −0.0126563
1.7 1.58216 9.98623 −5.49676 18.3203 15.7998 20.6771 −21.3541 72.7247 28.9856
1.8 2.62662 0.447567 −1.10085 9.81562 1.17559 −21.1571 −23.9045 −26.7997 25.7819
1.9 4.10090 −1.59161 8.81740 −13.7296 −6.52702 19.9578 3.35210 −24.4668 −56.3039
1.10 4.58446 −7.60062 13.0173 20.5295 −34.8447 −7.60859 23.0015 30.7694 94.1165
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(17\) \( -1 \)

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 2057.4.a.h 10
11.b odd 2 1 187.4.a.e ✓ 10
33.d even 2 1 1683.4.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.a.e ✓ 10 11.b odd 2 1
1683.4.a.j 10 33.d even 2 1
2057.4.a.h 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{10} + 4 T_{2}^{9} - 52 T_{2}^{8} - 180 T_{2}^{7} + 933 T_{2}^{6} + 2524 T_{2}^{5} - 6654 T_{2}^{4} + \cdots - 88 \) Copy content Toggle raw display
\( T_{3}^{10} - 11 T_{3}^{9} - 128 T_{3}^{8} + 1395 T_{3}^{7} + 5823 T_{3}^{6} - 56487 T_{3}^{5} + \cdots + 52224 \) Copy content Toggle raw display
\( T_{5}^{10} - 47 T_{5}^{9} + 184 T_{5}^{8} + 18893 T_{5}^{7} - 211549 T_{5}^{6} - 1945443 T_{5}^{5} + \cdots + 3660197440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + \cdots - 88 \) Copy content Toggle raw display
$3$ \( T^{10} - 11 T^{9} + \cdots + 52224 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 3660197440 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 2245822227552 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T - 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 37\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 23\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 76\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 50\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 75\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 92\!\cdots\!06 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
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