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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [73,4,Mod(1,73)] 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("73.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(73, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 73 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 73.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] 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: \(4.30713943042\)
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: \( x^{10} - 3 x^{9} - 57 x^{8} + 122 x^{7} + 1131 x^{6} - 1527 x^{5} - 9107 x^{4} + 6240 x^{3} + \cdots - 12288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{7} + 1) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{9} - \beta_{6} + \beta_{3} + \cdots + 3) q^{5} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots + 1) q^{6}+ \cdots + (25 \beta_{9} + 33 \beta_{8} + \cdots - 190) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 7 q^{2} + 10 q^{3} + 47 q^{4} + 24 q^{5} + 14 q^{6} + 42 q^{7} + 84 q^{8} + 156 q^{9} + 11 q^{10} + 74 q^{11} + 21 q^{12} - 2 q^{13} + 155 q^{14} + 120 q^{15} + 275 q^{16} + 84 q^{17} + 142 q^{18}+ \cdots - 1768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 57 x^{8} + 122 x^{7} + 1131 x^{6} - 1527 x^{5} - 9107 x^{4} + 6240 x^{3} + \cdots - 12288 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33 \nu^{9} - 565 \nu^{8} + 2329 \nu^{7} + 32366 \nu^{6} - 32995 \nu^{5} - 530793 \nu^{4} + \cdots + 246272 ) / 315904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27 \nu^{9} - 911 \nu^{8} + 2803 \nu^{7} + 56546 \nu^{6} - 69625 \nu^{5} - 1059811 \nu^{4} + \cdots - 5341184 ) / 315904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67 \nu^{9} - 573 \nu^{8} + 357 \nu^{7} + 13338 \nu^{6} - 88195 \nu^{5} + 53675 \nu^{4} + \cdots + 3051520 ) / 315904 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55 \nu^{9} - 119 \nu^{8} + 3059 \nu^{7} + 10342 \nu^{6} - 51701 \nu^{5} - 225699 \nu^{4} + \cdots - 1748224 ) / 157952 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{9} + 23 \nu^{8} - 1426 \nu^{7} - 1750 \nu^{6} + 30364 \nu^{5} + 37074 \nu^{4} + \cdots + 108928 ) / 39488 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 449 \nu^{9} - 2035 \nu^{8} - 23761 \nu^{7} + 89274 \nu^{6} + 454915 \nu^{5} - 1238751 \nu^{4} + \cdots - 2450944 ) / 315904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 369 \nu^{9} - 1535 \nu^{8} - 17741 \nu^{7} + 60566 \nu^{6} + 276955 \nu^{5} - 719619 \nu^{4} + \cdots - 1662464 ) / 157952 \) 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
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + 21\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 4\beta_{8} - 2\beta_{7} - 12\beta_{6} - 4\beta_{5} + 12\beta_{3} + 30\beta_{2} + 53\beta _1 + 252 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{9} - 28 \beta_{8} - 86 \beta_{7} - 108 \beta_{6} - 76 \beta_{5} - 32 \beta_{4} + 84 \beta_{3} + \cdots + 552 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 107 \beta_{9} - 181 \beta_{8} - 204 \beta_{7} - 654 \beta_{6} - 262 \beta_{5} + 7 \beta_{4} + \cdots + 6607 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 920 \beta_{9} - 888 \beta_{8} - 3160 \beta_{7} - 4600 \beta_{6} - 2624 \beta_{5} - 808 \beta_{4} + \cdots + 22496 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4336 \beta_{9} - 6632 \beta_{8} - 10960 \beta_{7} - 27960 \beta_{6} - 11936 \beta_{5} + 224 \beta_{4} + \cdots + 196628 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30625 \beta_{9} - 31465 \beta_{8} - 111602 \beta_{7} - 180682 \beta_{6} - 91778 \beta_{5} + \cdots + 854451 \) 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
5.99478
4.61559
3.13225
2.55471
0.748211
−0.745653
−1.93569
−2.85659
−3.93051
−4.57708
−4.99478 6.39333 16.9478 11.1263 −31.9333 −7.68924 −44.6925 13.8746 −55.5734
1.2 −3.61559 −3.31434 5.07246 −5.69861 11.9833 −18.4703 10.5848 −16.0151 20.6038
1.3 −2.13225 −8.06305 −3.45351 −12.7846 17.1925 8.57839 24.4217 38.0128 27.2599
1.4 −1.55471 5.94772 −5.58289 4.72645 −9.24696 20.5392 21.1174 8.37537 −7.34825
1.5 0.251789 −6.88442 −7.93660 13.5358 −1.73342 23.4434 −4.01266 20.3952 3.40818
1.6 1.74565 8.72107 −4.95270 18.1923 15.2240 −24.6039 −22.6109 49.0571 31.7574
1.7 2.93569 8.66231 0.618295 −12.3118 25.4299 32.2651 −21.6704 48.0357 −36.1436
1.8 3.85659 0.981034 6.87332 10.6723 3.78345 8.75399 −4.34515 −26.0376 41.1588
1.9 4.93051 4.74961 16.3099 −7.98655 23.4180 −20.3499 40.9722 −4.44116 −39.3778
1.10 5.57708 −7.19327 23.1039 4.52835 −40.1174 19.5332 84.2355 24.7431 25.2550
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(73\) \( +1 \)

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 73.4.a.c ✓ 10
3.b odd 2 1 657.4.a.e 10
4.b odd 2 1 1168.4.a.i 10
5.b even 2 1 1825.4.a.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
73.4.a.c ✓ 10 1.a even 1 1 trivial
657.4.a.e 10 3.b odd 2 1
1168.4.a.i 10 4.b odd 2 1
1825.4.a.e 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 7 T_{2}^{9} - 39 T_{2}^{8} + 322 T_{2}^{7} + 347 T_{2}^{6} - 4503 T_{2}^{5} + 335 T_{2}^{4} + \cdots + 8192 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(73))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 7 T^{9} + \cdots + 8192 \) Copy content Toggle raw display
$3$ \( T^{10} - 10 T^{9} + \cdots + 17713888 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 4483244504 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 1620487622772 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 3020344795808 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 157516393818008 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 56\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 47\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 38\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 30\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 27\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 73\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 23\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 76\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 55\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 10\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( (T + 73)^{10} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 30\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 61\!\cdots\!44 \) Copy content Toggle raw display
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