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Properties

Label 1870.4.a.j
Level $1870$
Weight $4$
Character orbit 1870.a
Self dual yes
Analytic conductor $110.334$
Analytic rank $1$
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] = [1870,4,Mod(1,1870)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1870, 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("1870.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1870 = 2 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1870.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-20,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(110.333571711\)
Analytic rank: \(1\)
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: \( x^{10} - x^{9} - 168 x^{8} + 138 x^{7} + 9025 x^{6} - 8357 x^{5} - 177203 x^{4} + 269951 x^{3} + \cdots + 1239820 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} - 5 q^{5} + 2 \beta_1 q^{6} - \beta_{7} q^{7} - 8 q^{8} + (\beta_{2} + 7) q^{9} + 10 q^{10} - 11 q^{11} - 4 \beta_1 q^{12} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{13}+ \cdots + ( - 11 \beta_{2} - 77) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{2} - q^{3} + 40 q^{4} - 50 q^{5} + 2 q^{6} + 4 q^{7} - 80 q^{8} + 67 q^{9} + 100 q^{10} - 110 q^{11} - 4 q^{12} - 7 q^{13} - 8 q^{14} + 5 q^{15} + 160 q^{16} + 170 q^{17} - 134 q^{18} + 11 q^{19}+ \cdots - 737 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 168 x^{8} + 138 x^{7} + 9025 x^{6} - 8357 x^{5} - 177203 x^{4} + 269951 x^{3} + \cdots + 1239820 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1418864 \nu^{9} + 14345391 \nu^{8} + 2139046692 \nu^{7} - 1995578838 \nu^{6} + \cdots + 61946558969024 ) / 2260973693961 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35618030 \nu^{9} - 40587027 \nu^{8} - 6041299572 \nu^{7} + 3797467365 \nu^{6} + \cdots - 38434968318281 ) / 753657897987 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 261476227 \nu^{9} + 644823840 \nu^{8} - 42017831958 \nu^{7} - 110257158024 \nu^{6} + \cdots + 26248026321743 ) / 2260973693961 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 120921722 \nu^{9} + 168498822 \nu^{8} - 19034162997 \nu^{7} - 32230000527 \nu^{6} + \cdots - 33614659373033 ) / 753657897987 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 401828813 \nu^{9} - 581928192 \nu^{8} + 66570823938 \nu^{7} + 101991030765 \nu^{6} + \cdots + 327314334748880 ) / 2260973693961 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 593618545 \nu^{9} - 1591189881 \nu^{8} + 99268261587 \nu^{7} + 267798251931 \nu^{6} + \cdots + 191533010569300 ) / 2260973693961 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1135239893 \nu^{9} - 908653584 \nu^{8} + 188074882305 \nu^{7} + 184311433641 \nu^{6} + \cdots + 792007981928909 ) / 2260973693961 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{7} - 3\beta_{6} + 3\beta_{5} + 6\beta_{4} + 4\beta_{3} + \beta_{2} + 63\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{9} - 3 \beta_{8} + 17 \beta_{7} - 4 \beta_{6} - \beta_{5} - 11 \beta_{4} + 6 \beta_{3} + \cdots + 2073 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 97 \beta_{9} - 33 \beta_{8} - 188 \beta_{7} - 298 \beta_{6} + 224 \beta_{5} + 610 \beta_{4} + \cdots + 1549 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 847 \beta_{9} - 321 \beta_{8} + 1772 \beta_{7} - 702 \beta_{6} - 126 \beta_{5} - 1413 \beta_{4} + \cdots + 149832 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7971 \beta_{9} - 4728 \beta_{8} - 14994 \beta_{7} - 25760 \beta_{6} + 15205 \beta_{5} + 52883 \beta_{4} + \cdots + 211329 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 78971 \beta_{9} - 32334 \beta_{8} + 151666 \beta_{7} - 86005 \beta_{6} - 10495 \beta_{5} + \cdots + 11637911 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 620371 \beta_{9} - 503403 \beta_{8} - 1143932 \beta_{7} - 2194644 \beta_{6} + 1062387 \beta_{5} + \cdots + 23098684 \) 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
9.42709
7.35577
3.72392
2.70822
1.93833
0.733468
−3.85082
−5.39267
−6.75882
−8.88449
−2.00000 −9.42709 4.00000 −5.00000 18.8542 7.39972 −8.00000 61.8701 10.0000
1.2 −2.00000 −7.35577 4.00000 −5.00000 14.7115 −8.63058 −8.00000 27.1073 10.0000
1.3 −2.00000 −3.72392 4.00000 −5.00000 7.44784 32.6645 −8.00000 −13.1324 10.0000
1.4 −2.00000 −2.70822 4.00000 −5.00000 5.41644 −12.8742 −8.00000 −19.6656 10.0000
1.5 −2.00000 −1.93833 4.00000 −5.00000 3.87666 17.2000 −8.00000 −23.2429 10.0000
1.6 −2.00000 −0.733468 4.00000 −5.00000 1.46694 −34.7632 −8.00000 −26.4620 10.0000
1.7 −2.00000 3.85082 4.00000 −5.00000 −7.70164 3.86013 −8.00000 −12.1712 10.0000
1.8 −2.00000 5.39267 4.00000 −5.00000 −10.7853 27.8251 −8.00000 2.08092 10.0000
1.9 −2.00000 6.75882 4.00000 −5.00000 −13.5176 −13.5453 −8.00000 18.6817 10.0000
1.10 −2.00000 8.88449 4.00000 −5.00000 −17.7690 −15.1361 −8.00000 51.9341 10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)
\(17\) \( -1 \)

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 1870.4.a.j ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.4.a.j ✓ 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + T_{3}^{9} - 168 T_{3}^{8} - 138 T_{3}^{7} + 9025 T_{3}^{6} + 8357 T_{3}^{5} + \cdots + 1239820 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1870))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 1239820 \) Copy content Toggle raw display
$5$ \( (T + 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 353627475520 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 261880959889958 \) Copy content Toggle raw display
$17$ \( (T - 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 15\!\cdots\!58 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 49\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 71\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 67\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 80\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 77\!\cdots\!50 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 66\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 92\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 28\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 30\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 21\!\cdots\!70 \) Copy content Toggle raw display
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