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Properties

Label 273.4.i.c
Level $273$
Weight $4$
Character orbit 273.i
Analytic conductor $16.108$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [273,4,Mod(79,273)] 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("273.79"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 273.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1075214316\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 32 x^{12} - 67 x^{11} + 852 x^{10} - 1355 x^{9} + 6603 x^{8} - 1490 x^{7} + \cdots + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{2} - 3 \beta_{4} q^{3} + ( - \beta_{7} - 2 \beta_{4} + \cdots - \beta_1) q^{4} + ( - \beta_{9} + 3 \beta_{4} - 3 \beta_{2} + \cdots - 3) q^{5} + ( - 3 \beta_{2} + 3) q^{6}+ \cdots + (9 \beta_{10} + 45 \beta_{8} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 8 q^{2} - 21 q^{3} - 16 q^{4} - 21 q^{5} + 48 q^{6} - 34 q^{7} + 36 q^{8} - 63 q^{9} + 150 q^{10} - 11 q^{11} - 48 q^{12} - 182 q^{13} + 74 q^{14} + 126 q^{15} - 116 q^{16} + 367 q^{17} - 72 q^{18}+ \cdots + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 32 x^{12} - 67 x^{11} + 852 x^{10} - 1355 x^{9} + 6603 x^{8} - 1490 x^{7} + \cdots + 6400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 222562902607249 \nu^{13} - 803852795374049 \nu^{12} + \cdots + 12\!\cdots\!40 ) / 39\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27594383858427 \nu^{13} + 11183470665505 \nu^{12} - 771354891728912 \nu^{11} + \cdots + 28\!\cdots\!48 ) / 30\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!37 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 96974980482941 \nu^{13} + 239087015123158 \nu^{12} + \cdots + 28\!\cdots\!00 ) / 27\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 380509617538091 \nu^{13} - 973817618000374 \nu^{12} + \cdots - 14\!\cdots\!96 ) / 90\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!13 \nu^{13} + \cdots + 88\!\cdots\!80 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!74 \nu^{13} + \cdots - 92\!\cdots\!84 ) / 36\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!63 \nu^{13} + \cdots - 21\!\cdots\!00 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!79 \nu^{13} + \cdots - 12\!\cdots\!44 ) / 39\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51\!\cdots\!07 \nu^{13} + \cdots - 17\!\cdots\!00 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 51\!\cdots\!31 \nu^{13} + \cdots - 91\!\cdots\!60 ) / 26\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!39 \nu^{13} + \cdots - 17\!\cdots\!40 ) / 39\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 9\beta_{4} - \beta_{2} - \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 2\beta_{5} - 3\beta_{3} + 21\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - 4\beta_{12} - 2\beta_{10} + 4\beta_{8} - 25\beta_{7} - 177\beta_{4} + 25\beta_{3} + 51\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} + 2 \beta_{12} + 56 \beta_{11} - 54 \beta_{9} + 103 \beta_{7} + 56 \beta_{6} + \cdots - 443 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58\beta_{10} - 116\beta_{8} - 44\beta_{6} - 48\beta_{5} - 669\beta_{3} + 1775\beta_{2} + 4293 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 102 \beta_{13} - 142 \beta_{12} - 1376 \beta_{11} - 102 \beta_{10} + 1310 \beta_{9} + 142 \beta_{8} + \cdots + 13491 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1478 \beta_{13} + 2920 \beta_{12} + 2428 \beta_{11} - 2520 \beta_{9} + 18529 \beta_{7} + \cdots - 112661 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3906\beta_{10} - 6242\beta_{8} - 34080\beta_{6} - 32454\beta_{5} - 91727\beta_{3} + 365995\beta_{2} + 475059 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 37986 \beta_{13} - 73692 \beta_{12} - 92988 \beta_{11} - 37986 \beta_{10} + 93904 \beta_{9} + \cdots + 1675567 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 130974 \beta_{13} + 223046 \beta_{12} + 873600 \beta_{11} - 836062 \beta_{9} + 2661451 \beta_{7} + \cdots - 14226491 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1004574 \beta_{10} - 1915712 \beta_{8} - 3093660 \beta_{6} - 3076664 \beta_{5} - 14752553 \beta_{3} + \cdots + 85116405 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4098234 \beta_{13} - 7208890 \beta_{12} - 23161408 \beta_{11} - 4098234 \beta_{10} + \cdots + 284181739 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
2.08566 + 3.61247i
1.35564 + 2.34804i
1.19776 + 2.07458i
−0.226672 − 0.392608i
−0.338439 − 0.586193i
−0.899523 − 1.55802i
−2.67443 − 4.63224i
2.08566 − 3.61247i
1.35564 − 2.34804i
1.19776 − 2.07458i
−0.226672 + 0.392608i
−0.338439 + 0.586193i
−0.899523 + 1.55802i
−2.67443 + 4.63224i
−2.58566 + 4.47850i −1.50000 − 2.59808i −9.37128 − 16.2315i 9.25468 − 16.0296i 15.5140 3.78596 + 18.1292i 55.5532 −4.50000 + 7.79423i 47.8589 + 82.8941i
79.2 −1.85564 + 3.21406i −1.50000 − 2.59808i −2.88680 − 5.00008i 4.29883 − 7.44580i 11.1338 −9.26332 − 16.0372i −8.26280 −4.50000 + 7.79423i 15.9542 + 27.6335i
79.3 −1.69776 + 2.94061i −1.50000 − 2.59808i −1.76479 − 3.05670i −6.79542 + 11.7700i 10.1866 −10.4073 + 15.3195i −15.1794 −4.50000 + 7.79423i −23.0740 − 39.9654i
79.4 −0.273328 + 0.473418i −1.50000 − 2.59808i 3.85058 + 6.66941i 0.100649 − 0.174329i 1.63997 −10.0223 − 15.5741i −8.58313 −4.50000 + 7.79423i 0.0550204 + 0.0952982i
79.5 −0.161561 + 0.279832i −1.50000 − 2.59808i 3.94780 + 6.83778i −4.12339 + 7.14192i 0.969367 −14.9851 + 10.8833i −5.13622 −4.50000 + 7.79423i −1.33236 − 2.30771i
79.6 0.399523 − 0.691995i −1.50000 − 2.59808i 3.68076 + 6.37527i −6.20326 + 10.7444i −2.39714 13.6240 − 12.5454i 12.2746 −4.50000 + 7.79423i 4.95669 + 8.58524i
79.7 2.17443 − 3.76622i −1.50000 − 2.59808i −5.45627 − 9.45054i −7.03210 + 12.1799i −13.0466 10.2681 + 15.4132i −12.6662 −4.50000 + 7.79423i 30.5816 + 52.9688i
235.1 −2.58566 − 4.47850i −1.50000 + 2.59808i −9.37128 + 16.2315i 9.25468 + 16.0296i 15.5140 3.78596 − 18.1292i 55.5532 −4.50000 − 7.79423i 47.8589 − 82.8941i
235.2 −1.85564 − 3.21406i −1.50000 + 2.59808i −2.88680 + 5.00008i 4.29883 + 7.44580i 11.1338 −9.26332 + 16.0372i −8.26280 −4.50000 − 7.79423i 15.9542 − 27.6335i
235.3 −1.69776 − 2.94061i −1.50000 + 2.59808i −1.76479 + 3.05670i −6.79542 − 11.7700i 10.1866 −10.4073 − 15.3195i −15.1794 −4.50000 − 7.79423i −23.0740 + 39.9654i
235.4 −0.273328 − 0.473418i −1.50000 + 2.59808i 3.85058 − 6.66941i 0.100649 + 0.174329i 1.63997 −10.0223 + 15.5741i −8.58313 −4.50000 − 7.79423i 0.0550204 − 0.0952982i
235.5 −0.161561 − 0.279832i −1.50000 + 2.59808i 3.94780 − 6.83778i −4.12339 − 7.14192i 0.969367 −14.9851 − 10.8833i −5.13622 −4.50000 − 7.79423i −1.33236 + 2.30771i
235.6 0.399523 + 0.691995i −1.50000 + 2.59808i 3.68076 − 6.37527i −6.20326 − 10.7444i −2.39714 13.6240 + 12.5454i 12.2746 −4.50000 − 7.79423i 4.95669 − 8.58524i
235.7 2.17443 + 3.76622i −1.50000 + 2.59808i −5.45627 + 9.45054i −7.03210 − 12.1799i −13.0466 10.2681 − 15.4132i −12.6662 −4.50000 − 7.79423i 30.5816 − 52.9688i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.4.i.c ✓ 14
7.c even 3 1 inner 273.4.i.c ✓ 14
7.c even 3 1 1911.4.a.u 7
7.d odd 6 1 1911.4.a.t 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.4.i.c ✓ 14 1.a even 1 1 trivial
273.4.i.c ✓ 14 7.c even 3 1 inner
1911.4.a.t 7 7.d odd 6 1
1911.4.a.u 7 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 8 T_{2}^{13} + 68 T_{2}^{12} + 276 T_{2}^{11} + 1541 T_{2}^{10} + 5362 T_{2}^{9} + \cdots + 1600 \) acting on \(S_{4}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 8 T^{13} + \cdots + 1600 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 392477190400 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 55\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 96\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( (T + 13)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 41\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 16\!\cdots\!41)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 42\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 492448628815280)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 76\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 11\!\cdots\!63)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 53\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 11\!\cdots\!60)^{2} \) Copy content Toggle raw display
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