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Properties

Label 279.3.v.a
Level $279$
Weight $3$
Character orbit 279.v
Analytic conductor $7.602$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [279,3,Mod(46,279)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(279, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("279.46"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 279.v (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,3,0,-11,14,0,-1,19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.60219937565\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 20 x^{18} - 33 x^{17} + 250 x^{16} - 510 x^{15} + 2908 x^{14} - 6447 x^{13} + \cdots + 731025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + (\beta_{19} + 3 \beta_{11}) q^{4} + (\beta_{12} - \beta_{8} + \beta_{6} + \cdots + 2) q^{5} + ( - \beta_{19} + \beta_{15} + \cdots - \beta_{7}) q^{7} + (2 \beta_{18} + \beta_{17} + \beta_{16} + \cdots + 2) q^{8}+ \cdots + ( - 5 \beta_{19} - 5 \beta_{17} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} - 11 q^{4} + 14 q^{5} - q^{7} + 19 q^{8} + 12 q^{10} + 10 q^{11} + 10 q^{13} - 103 q^{16} - 35 q^{17} + 47 q^{19} + 125 q^{20} + 150 q^{22} - 75 q^{23} + 82 q^{25} + 88 q^{28} - 5 q^{29}+ \cdots + 1000 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 20 x^{18} - 33 x^{17} + 250 x^{16} - 510 x^{15} + 2908 x^{14} - 6447 x^{13} + \cdots + 731025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44\!\cdots\!91 \nu^{19} + \cdots + 79\!\cdots\!35 ) / 11\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!64 \nu^{19} + \cdots + 84\!\cdots\!50 ) / 11\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!28 \nu^{19} + \cdots + 43\!\cdots\!50 ) / 10\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!83 \nu^{19} + \cdots + 58\!\cdots\!50 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!66 \nu^{19} + \cdots - 58\!\cdots\!50 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!70 \nu^{19} + \cdots + 34\!\cdots\!50 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!82 \nu^{19} + \cdots + 32\!\cdots\!80 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!77 \nu^{19} + \cdots - 17\!\cdots\!25 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 78\!\cdots\!61 \nu^{19} + \cdots - 47\!\cdots\!50 ) / 10\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!98 \nu^{19} + \cdots - 10\!\cdots\!90 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 80\!\cdots\!48 \nu^{19} + \cdots - 27\!\cdots\!50 ) / 64\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!83 \nu^{19} + \cdots + 67\!\cdots\!75 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!13 \nu^{19} + \cdots - 12\!\cdots\!75 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!58 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 23\!\cdots\!90 \nu^{19} + \cdots + 23\!\cdots\!50 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 24\!\cdots\!27 \nu^{19} + \cdots + 21\!\cdots\!35 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 59\!\cdots\!49 \nu^{19} + \cdots + 86\!\cdots\!40 ) / 12\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 42\!\cdots\!56 \nu^{19} + \cdots + 69\!\cdots\!30 ) / 58\!\cdots\!65 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} - 7\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{18} + \beta_{17} + 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{10} + 2 \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{19} - \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 69 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{19} - 19 \beta_{17} - 32 \beta_{15} + 32 \beta_{14} + 36 \beta_{13} + 2 \beta_{12} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{18} + 164 \beta_{17} + 22 \beta_{16} - 6 \beta_{15} - 32 \beta_{14} - 34 \beta_{13} + \cdots + 184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 280 \beta_{19} - 444 \beta_{18} + 64 \beta_{16} - 50 \beta_{14} - 134 \beta_{13} - 444 \beta_{12} + \cdots + 190 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1719 \beta_{19} + 570 \beta_{18} - 2109 \beta_{17} - 1719 \beta_{16} + 158 \beta_{15} - 570 \beta_{14} + \cdots - 11833 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1008 \beta_{19} + 6924 \beta_{18} + 3841 \beta_{17} + 3841 \beta_{16} + 5998 \beta_{15} + 1190 \beta_{13} + \cdots + 6494 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6263 \beta_{19} + 6263 \beta_{17} - 8016 \beta_{15} + 8016 \beta_{14} + 16328 \beta_{13} - 4938 \beta_{12} + \cdots + 156379 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 95984 \beta_{18} - 37104 \beta_{17} - 51187 \beta_{16} - 95984 \beta_{15} + 15308 \beta_{14} + \cdots - 145498 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 95140 \beta_{19} - 110600 \beta_{18} + 266540 \beta_{16} + 56836 \beta_{14} - 130108 \beta_{13} + \cdots - 871692 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 184856 \beta_{19} + 1085696 \beta_{18} + 487544 \beta_{17} - 184856 \beta_{16} + 1325416 \beta_{15} + \cdots + 227080 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3381373 \beta_{19} + 888180 \beta_{18} - 1398220 \beta_{17} - 1398220 \beta_{16} + 1519456 \beta_{15} + \cdots + 1910816 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 8761393 \beta_{19} - 8761393 \beta_{17} - 14631402 \beta_{15} + 14631402 \beta_{14} + 18020750 \beta_{13} + \cdots + 3398490 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 14172340 \beta_{18} + 63366093 \beta_{17} + 20130213 \beta_{16} - 14172340 \beta_{15} + \cdots + 146888192 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 113596275 \beta_{19} - 197466448 \beta_{18} + 28815675 \beta_{16} - 54361786 \beta_{14} + \cdots + 35190172 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 556031402 \beta_{19} + 287901398 \beta_{18} - 841992700 \beta_{17} - 556031402 \beta_{16} + \cdots - 4994265487 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 348933632 \beta_{19} + 3469253478 \beta_{18} + 1468094680 \beta_{17} + 1468094680 \beta_{16} + \cdots + 2873523836 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−1.12738 − 3.46973i
−0.453363 − 1.39531i
0.276623 + 0.851359i
0.651822 + 2.00610i
1.15230 + 3.54642i
−1.12738 + 3.46973i
−0.453363 + 1.39531i
0.276623 − 0.851359i
0.651822 − 2.00610i
1.15230 − 3.54642i
−2.86150 − 2.07900i
−1.55178 − 1.12743i
0.339254 + 0.246482i
2.18177 + 1.58515i
2.39225 + 1.73807i
−2.86150 + 2.07900i
−1.55178 + 1.12743i
0.339254 − 0.246482i
2.18177 − 1.58515i
2.39225 − 1.73807i
−2.95153 + 2.14441i 0 2.87696 − 8.85438i 3.94423 0 −0.932951 + 2.87133i 5.98647 + 18.4245i 0 −11.6415 + 8.45807i
46.2 −1.18692 + 0.862348i 0 −0.570933 + 1.75715i 4.14776 0 1.06697 − 3.28380i −2.65108 − 8.15917i 0 −4.92306 + 3.57681i
46.3 0.724209 − 0.526168i 0 −0.988443 + 3.04211i −4.26245 0 1.06330 − 3.27251i 1.99132 + 6.12865i 0 −3.08690 + 2.24277i
46.4 1.70649 − 1.23984i 0 0.138848 − 0.427329i −4.35092 0 −2.01919 + 6.21441i 2.31441 + 7.12303i 0 −7.42481 + 5.39444i
46.5 3.01677 − 2.19181i 0 3.06079 − 9.42013i 7.37547 0 −1.10519 + 3.40142i −6.80424 − 20.9413i 0 22.2501 − 16.1656i
91.1 −2.95153 − 2.14441i 0 2.87696 + 8.85438i 3.94423 0 −0.932951 − 2.87133i 5.98647 − 18.4245i 0 −11.6415 − 8.45807i
91.2 −1.18692 − 0.862348i 0 −0.570933 − 1.75715i 4.14776 0 1.06697 + 3.28380i −2.65108 + 8.15917i 0 −4.92306 − 3.57681i
91.3 0.724209 + 0.526168i 0 −0.988443 − 3.04211i −4.26245 0 1.06330 + 3.27251i 1.99132 − 6.12865i 0 −3.08690 − 2.24277i
91.4 1.70649 + 1.23984i 0 0.138848 + 0.427329i −4.35092 0 −2.01919 − 6.21441i 2.31441 − 7.12303i 0 −7.42481 − 5.39444i
91.5 3.01677 + 2.19181i 0 3.06079 + 9.42013i 7.37547 0 −1.10519 − 3.40142i −6.80424 + 20.9413i 0 22.2501 + 16.1656i
244.1 −1.09299 − 3.36389i 0 −6.88507 + 5.00229i −8.82683 0 2.08188 − 1.51257i 12.9065 + 9.37714i 0 9.64768 + 29.6925i
244.2 −0.592727 − 1.82423i 0 0.259595 − 0.188607i 6.67828 0 3.63065 − 2.63783i −6.70505 − 4.87150i 0 −3.95840 − 12.1827i
244.3 0.129583 + 0.398816i 0 3.09381 − 2.24778i −3.98636 0 −9.49164 + 6.89608i 2.65437 + 1.92851i 0 −0.516566 − 1.58983i
244.4 0.833363 + 2.56483i 0 −2.64778 + 1.92372i 1.06496 0 10.2892 − 7.47554i 1.58651 + 1.15266i 0 0.887497 + 2.73143i
244.5 0.913758 + 2.81226i 0 −3.83778 + 2.78831i 5.21585 0 −5.08304 + 3.69304i −1.77924 − 1.29269i 0 4.76602 + 14.6683i
271.1 −1.09299 + 3.36389i 0 −6.88507 − 5.00229i −8.82683 0 2.08188 + 1.51257i 12.9065 − 9.37714i 0 9.64768 − 29.6925i
271.2 −0.592727 + 1.82423i 0 0.259595 + 0.188607i 6.67828 0 3.63065 + 2.63783i −6.70505 + 4.87150i 0 −3.95840 + 12.1827i
271.3 0.129583 − 0.398816i 0 3.09381 + 2.24778i −3.98636 0 −9.49164 − 6.89608i 2.65437 − 1.92851i 0 −0.516566 + 1.58983i
271.4 0.833363 − 2.56483i 0 −2.64778 − 1.92372i 1.06496 0 10.2892 + 7.47554i 1.58651 − 1.15266i 0 0.887497 − 2.73143i
271.5 0.913758 − 2.81226i 0 −3.83778 − 2.78831i 5.21585 0 −5.08304 − 3.69304i −1.77924 + 1.29269i 0 4.76602 − 14.6683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 279.3.v.a 20
3.b odd 2 1 31.3.f.a ✓ 20
31.f odd 10 1 inner 279.3.v.a 20
93.k even 10 1 31.3.f.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.3.f.a ✓ 20 3.b odd 2 1
31.3.f.a ✓ 20 93.k even 10 1
279.3.v.a 20 1.a even 1 1 trivial
279.3.v.a 20 31.f odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 3 T_{2}^{19} + 20 T_{2}^{18} - 52 T_{2}^{17} + 310 T_{2}^{16} - 665 T_{2}^{15} + \cdots + 731025 \) acting on \(S_{3}^{\mathrm{new}}(279, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 731025 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 7 T^{9} + \cdots + 2920851)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 82363701948025 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 70\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
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