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Properties

Label 19.10.a.b
Level $19$
Weight $10$
Character orbit 19.a
Self dual yes
Analytic conductor $9.786$
Analytic rank $0$
Dimension $8$
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] = [19,10,Mod(1,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 19.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.78568088711\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} + \beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 331) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 490) q^{5} + (2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots - 1207) q^{6}+ \cdots + ( - 187353 \beta_{7} + \cdots - 211047948) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 15 q^{2} + 7 q^{3} + 2645 q^{4} + 3894 q^{5} - 9723 q^{6} - 7133 q^{7} + 10911 q^{8} + 102715 q^{9} + 113172 q^{10} + 172818 q^{11} + 349117 q^{12} + 109291 q^{13} + 250959 q^{14} + 457332 q^{15}+ \cdots - 1682553420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2442607 \nu^{7} + 89863001 \nu^{6} + 5717983110 \nu^{5} - 218132022350 \nu^{4} + \cdots + 25\!\cdots\!72 ) / 268007397120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2442607 \nu^{7} + 89863001 \nu^{6} + 5717983110 \nu^{5} - 218132022350 \nu^{4} + \cdots + 27\!\cdots\!52 ) / 268007397120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4307827 \nu^{7} - 91728929 \nu^{6} - 11718846762 \nu^{5} + 209079010526 \nu^{4} + \cdots + 19\!\cdots\!04 ) / 268007397120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2957621 \nu^{7} - 103632415 \nu^{6} - 7003343598 \nu^{5} + 249515611234 \nu^{4} + \cdots - 34\!\cdots\!12 ) / 89335799040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10018537 \nu^{7} - 278554463 \nu^{6} - 25382046138 \nu^{5} + 653708092034 \nu^{4} + \cdots - 52\!\cdots\!88 ) / 268007397120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3205844 \nu^{7} - 67957249 \nu^{6} - 8652761913 \nu^{5} + 153606755494 \nu^{4} + \cdots + 634795682678720 ) / 67001849280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 2\beta _1 + 839 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{7} - 5\beta_{6} - \beta_{5} - 8\beta_{4} - 16\beta_{3} + 4\beta_{2} + 1457\beta _1 + 1575 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -93\beta_{7} + 257\beta_{6} - 301\beta_{5} + 2\beta_{4} - 2404\beta_{3} + 1880\beta_{2} + 10395\beta _1 + 1224153 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11883 \beta_{7} - 8443 \beta_{6} - 7181 \beta_{5} - 22534 \beta_{4} - 46878 \beta_{3} + 8806 \beta_{2} + \cdots + 8671551 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 235743 \beta_{7} + 751431 \beta_{6} - 911307 \beta_{5} - 91222 \beta_{4} - 5100850 \beta_{3} + \cdots + 2014920109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23484551 \beta_{7} - 11105367 \beta_{6} - 22663853 \beta_{5} - 49941242 \beta_{4} - 115934890 \beta_{3} + \cdots + 23420404735 \) 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
45.0760
34.1433
8.83232
6.59102
5.24681
−28.7385
−28.7538
−41.3971
−43.0760 210.029 1343.54 1106.01 −9047.23 1237.92 −35819.6 24429.4 −47642.6
1.2 −32.1433 6.24328 521.193 −1240.15 −200.680 −8874.26 −295.489 −19644.0 39862.5
1.3 −6.83232 −260.789 −465.319 −1115.83 1781.79 −8602.80 6677.36 48328.0 7623.74
1.4 −4.59102 196.638 −490.923 2263.73 −902.769 −2177.39 4604.44 18983.5 −10392.8
1.5 −3.24681 −81.1272 −501.458 −574.796 263.404 7689.68 3290.50 −13101.4 1866.25
1.6 30.7385 −266.878 432.858 2199.99 −8203.46 7318.17 −2432.70 51541.1 67624.5
1.7 30.7538 175.481 433.799 18.5839 5396.70 6967.83 −2404.99 11110.5 571.527
1.8 43.3971 27.4035 1371.31 1236.46 1189.23 −10692.2 37291.4 −18932.0 53658.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \( -1 \)

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 19.10.a.b ✓ 8
3.b odd 2 1 171.10.a.f 8
4.b odd 2 1 304.10.a.i 8
19.b odd 2 1 361.10.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.10.a.b ✓ 8 1.a even 1 1 trivial
171.10.a.f 8 3.b odd 2 1
304.10.a.i 8 4.b odd 2 1
361.10.a.c 8 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 15 T_{2}^{7} - 3258 T_{2}^{6} + 41238 T_{2}^{5} + 2972568 T_{2}^{4} - 23100984 T_{2}^{3} + \cdots - 5784998400 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(19))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots - 5784998400 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots - 70\!\cdots\!40 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 47\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 41\!\cdots\!34 \) Copy content Toggle raw display
$19$ \( (T - 130321)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 95\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 55\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 16\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 56\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 39\!\cdots\!66 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 50\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
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