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Properties

Label 79.4.a.c
Level $79$
Weight $4$
Character orbit 79.a
Self dual yes
Analytic conductor $4.661$
Analytic rank $0$
Dimension $12$
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] = [79,4,Mod(1,79)] 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("79.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(79, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 79.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(4.66115089045\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 68 x^{10} + 262 x^{9} + 1631 x^{8} - 5739 x^{7} - 17428 x^{6} + 48800 x^{5} + \cdots + 6224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{9} + 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots + 3) q^{5} + ( - \beta_{10} - 3 \beta_{9} + \cdots + 2) q^{6} + (\beta_{10} + \beta_{9} + \cdots + 2 \beta_1) q^{7}+ \cdots + (67 \beta_{11} + 39 \beta_{10} + \cdots - 743) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 8 q^{3} + 56 q^{4} + 39 q^{5} + 21 q^{6} + 16 q^{7} + 30 q^{8} + 204 q^{9} + 25 q^{10} + 91 q^{11} + 127 q^{12} + 57 q^{13} + 177 q^{14} + 86 q^{15} + 260 q^{16} + 338 q^{17} + 265 q^{18}+ \cdots - 8831 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 68 x^{10} + 262 x^{9} + 1631 x^{8} - 5739 x^{7} - 17428 x^{6} + 48800 x^{5} + \cdots + 6224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 850307337 \nu^{11} + 5527629291 \nu^{10} - 59714119623 \nu^{9} - 289980230827 \nu^{8} + \cdots - 473501514607280 ) / 129171135312640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3685871483 \nu^{11} + 14114359439 \nu^{10} + 199158097013 \nu^{9} + \cdots - 567992420625136 ) / 64585567656320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 395551436 \nu^{11} - 1861761903 \nu^{10} - 23950179850 \nu^{9} + 116133583745 \nu^{8} + \cdots - 41991904325420 ) / 2018298989260 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36038584685 \nu^{11} - 182367912681 \nu^{10} - 2186687175203 \nu^{9} + \cdots - 57879722101616 ) / 129171135312640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19810049281 \nu^{11} + 12521022445 \nu^{10} + 1469629646543 \nu^{9} + \cdots + 382715058415408 ) / 64585567656320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21162088125 \nu^{11} - 18571215249 \nu^{10} - 1476612964867 \nu^{9} + \cdots + 404001759374096 ) / 64585567656320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 44620308289 \nu^{11} + 146677514477 \nu^{10} + 2979694567567 \nu^{9} + \cdots - 274592608875984 ) / 129171135312640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12656915861 \nu^{11} - 58201885669 \nu^{10} - 899174938723 \nu^{9} + 3921224878469 \nu^{8} + \cdots + 226197004203344 ) / 32292783828160 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 51287712611 \nu^{11} - 274594603879 \nu^{10} - 3137056452557 \nu^{9} + \cdots - 71693315005328 ) / 129171135312640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 3\beta_{9} + \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + \beta_{10} + 7 \beta_{9} + \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 7 \beta_{5} + \cdots + 274 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} + 35 \beta_{10} + 116 \beta_{9} + 31 \beta_{8} - 9 \beta_{7} - 7 \beta_{6} + 33 \beta_{5} + \cdots - 99 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58 \beta_{11} + 33 \beta_{10} + 270 \beta_{9} + 21 \beta_{8} - 94 \beta_{7} + 248 \beta_{6} + \cdots + 6571 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 525 \beta_{11} + 1029 \beta_{10} + 3566 \beta_{9} + 867 \beta_{8} - 380 \beta_{7} - 491 \beta_{6} + \cdots - 5054 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1206 \beta_{11} + 848 \beta_{10} + 7891 \beta_{9} + 188 \beta_{8} - 3146 \beta_{7} + 9066 \beta_{6} + \cdots + 168159 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19819 \beta_{11} + 28866 \beta_{10} + 101829 \beta_{9} + 24400 \beta_{8} - 11334 \beta_{7} + \cdots - 192949 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 18352 \beta_{11} + 18750 \beta_{10} + 206876 \beta_{9} - 4906 \beta_{8} - 92387 \beta_{7} + \cdots + 4472920 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 658815 \beta_{11} + 798230 \beta_{10} + 2824536 \beta_{9} + 697568 \beta_{8} - 295070 \beta_{7} + \cdots - 6646756 \) 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
−5.43385
−4.49318
−3.06708
−1.97698
−1.41788
−0.166742
0.179727
2.68514
3.61815
4.13671
4.71954
5.21644
−5.43385 6.19432 21.5268 18.7373 −33.6590 −4.22067 −73.5024 11.3696 −101.816
1.2 −4.49318 −4.09992 12.1886 −7.80558 18.4217 −36.3907 −18.8202 −10.1907 35.0718
1.3 −3.06708 5.03337 1.40700 5.04763 −15.4378 5.57431 20.2213 −1.66516 −15.4815
1.4 −1.97698 −4.02409 −4.09156 −14.8508 7.95554 26.7944 23.9047 −10.8067 29.3598
1.5 −1.41788 −10.1539 −5.98962 −4.32107 14.3969 −11.7958 19.8356 76.1010 6.12675
1.6 −0.166742 8.69498 −7.97220 4.01731 −1.44982 19.9529 2.66324 48.6026 −0.669856
1.7 0.179727 −5.02563 −7.96770 15.3970 −0.903243 13.2627 −2.86983 −1.74304 2.76726
1.8 2.68514 6.43768 −0.790042 21.9463 17.2860 −28.1756 −23.6025 14.4437 58.9289
1.9 3.61815 3.24285 5.09102 3.27956 11.7331 21.1104 −10.5251 −16.4839 11.8660
1.10 4.13671 9.82638 9.11240 −17.4756 40.6489 −6.76855 4.60168 69.5578 −72.2915
1.11 4.71954 −8.84834 14.2740 14.5981 −41.7601 29.7255 29.6105 51.2931 68.8961
1.12 5.21644 0.722266 19.2113 0.429827 3.76766 −13.0689 58.4830 −26.4783 2.24217
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(79\) \( +1 \)

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 79.4.a.c ✓ 12
3.b odd 2 1 711.4.a.f 12
4.b odd 2 1 1264.4.a.m 12
5.b even 2 1 1975.4.a.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.4.a.c ✓ 12 1.a even 1 1 trivial
711.4.a.f 12 3.b odd 2 1
1264.4.a.m 12 4.b odd 2 1
1975.4.a.c 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4 T_{2}^{11} - 68 T_{2}^{10} + 262 T_{2}^{9} + 1631 T_{2}^{8} - 5739 T_{2}^{7} + \cdots + 6224 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(79))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots + 6224 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots - 299223424 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 23126692692 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 111997352538112 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 29\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 27\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 93\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 81\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 46\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 57\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 30\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( (T + 79)^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 35\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
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