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Properties

Label 84.12.i.a
Level $84$
Weight $12$
Character orbit 84.i
Analytic conductor $64.541$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.5408271670\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 6 x^{13} + 198245134 x^{12} + 414863096508 x^{11} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{12}\cdot 7^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + (243 \beta_1 + 243) q^{3} + ( - \beta_{3} - 1031 \beta_1) q^{5} + (\beta_{7} + 336 \beta_1 + 2668) q^{7} + 59049 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (243 \beta_1 + 243) q^{3} + ( - \beta_{3} - 1031 \beta_1) q^{5} + (\beta_{7} + 336 \beta_1 + 2668) q^{7} + 59049 \beta_1 q^{9} + ( - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3} - 13 \beta_{2} + 7776 \beta_1 + 7777) q^{11} + (\beta_{12} - \beta_{10} + \beta_{9} - 6 \beta_{7} + \beta_{6} - 4 \beta_{5} + \beta_{4} + \cdots + 109646) q^{13}+ \cdots + ( - 59049 \beta_{7} + 59049 \beta_{4} + 708588 \beta_{3} + \cdots - 459165024) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 1701 q^{3} + 7218 q^{5} + 35001 q^{7} - 413343 q^{9} + 54450 q^{11} + 1534982 q^{13} + 3507948 q^{15} + 1478880 q^{17} - 22875935 q^{19} + 3394224 q^{21} + 62540568 q^{23} - 62136141 q^{25} - 200884698 q^{27} + 102097728 q^{29} + 188600405 q^{31} - 13231350 q^{33} - 253840734 q^{35} + 199685599 q^{37} + 186500313 q^{39} - 693868716 q^{41} - 620701754 q^{43} + 426215682 q^{45} + 2771987346 q^{47} - 5209147075 q^{49} - 359367840 q^{51} + 6487034184 q^{53} + 10046238656 q^{55} - 11117704410 q^{57} - 8183838888 q^{59} + 4069556330 q^{61} - 1241977617 q^{63} - 1520229906 q^{65} + 15766443531 q^{67} + 30394716048 q^{69} - 33183285444 q^{71} - 31685143839 q^{73} + 15099082263 q^{75} + 3261253500 q^{77} + 21999509987 q^{79} - 24407490807 q^{81} - 63053885988 q^{83} + 35204204624 q^{85} + 12404873952 q^{87} + 67041904680 q^{89} - 190876959523 q^{91} - 45829898415 q^{93} + 133488871470 q^{95} + 284083418100 q^{97} - 6430436100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} + 198245134 x^{12} + 414863096508 x^{11} + \cdots + 37\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 50\!\cdots\!47 \nu^{13} + \cdots - 56\!\cdots\!12 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!47 \nu^{13} + \cdots - 82\!\cdots\!24 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 50\!\cdots\!66 \nu^{13} + \cdots - 19\!\cdots\!44 ) / 64\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 84\!\cdots\!04 \nu^{13} + \cdots + 70\!\cdots\!36 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!94 \nu^{13} + \cdots - 55\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73\!\cdots\!44 \nu^{13} + \cdots - 98\!\cdots\!64 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!62 \nu^{13} + \cdots - 65\!\cdots\!52 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!63 \nu^{13} + \cdots + 42\!\cdots\!28 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!31 \nu^{13} + \cdots - 20\!\cdots\!28 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\!\cdots\!99 \nu^{13} + \cdots + 23\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!09 \nu^{13} + \cdots - 38\!\cdots\!72 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{13} + \cdots - 43\!\cdots\!36 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 94\!\cdots\!21 \nu^{13} + \cdots + 95\!\cdots\!36 ) / 73\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11 \beta_{13} + 11 \beta_{12} - 38 \beta_{11} + 58 \beta_{9} - 20 \beta_{8} - 288 \beta_{7} - 38 \beta_{6} + 28 \beta_{5} - 1611 \beta_{3} + 86 \beta_{2} + 56641807 \beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 9727 \beta_{12} - 131125 \beta_{10} + 541963 \beta_{9} + 720314 \beta_{7} - 1727168 \beta_{6} - 309559 \beta_{5} - 124381 \beta_{4} - 74427061 \beta_{3} - 74140604 \beta_{2} + \cdots - 89165142443 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 1192000460 \beta_{13} + 3652634255 \beta_{11} - 2346232175 \beta_{10} + 9942804330 \beta_{9} + 2346232175 \beta_{8} + 47065322810 \beta_{7} + \cdots - 41\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1426989509488 \beta_{13} + 1426989509488 \beta_{12} - 7368234223504 \beta_{11} - 16799038997286 \beta_{9} + 20861313992065 \beta_{8} + \cdots - 9430804773782 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 97\!\cdots\!23 \beta_{12} + \cdots + 34\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 18\!\cdots\!89 \beta_{13} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 70\!\cdots\!94 \beta_{13} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 22\!\cdots\!90 \beta_{12} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 46\!\cdots\!93 \beta_{13} + \cdots - 27\!\cdots\!57 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 27\!\cdots\!75 \beta_{13} + \cdots + 91\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 26\!\cdots\!88 \beta_{12} + \cdots + 25\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 31\!\cdots\!64 \beta_{13} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
5091.70 − 8819.09i
4461.47 − 7727.49i
2987.06 − 5173.74i
−1534.38 + 2657.62i
−2677.56 + 4637.66i
−3853.65 + 6674.72i
−4471.65 + 7745.12i
5091.70 + 8819.09i
4461.47 + 7727.49i
2987.06 + 5173.74i
−1534.38 − 2657.62i
−2677.56 − 4637.66i
−3853.65 − 6674.72i
−4471.65 − 7745.12i
0 121.500 − 210.444i 0 −4575.70 − 7925.35i 0 −7241.01 + 43873.6i 0 −29524.5 − 51137.9i 0
25.2 0 121.500 − 210.444i 0 −3945.47 − 6833.75i 0 39151.4 − 21082.9i 0 −29524.5 − 51137.9i 0
25.3 0 121.500 − 210.444i 0 −2471.06 − 4280.00i 0 −15790.8 − 41569.0i 0 −29524.5 − 51137.9i 0
25.4 0 121.500 − 210.444i 0 2050.38 + 3551.36i 0 −44391.9 + 2586.27i 0 −29524.5 − 51137.9i 0
25.5 0 121.500 − 210.444i 0 3193.56 + 5531.40i 0 42025.8 + 14531.2i 0 −29524.5 − 51137.9i 0
25.6 0 121.500 − 210.444i 0 4369.65 + 7568.45i 0 −2593.70 − 44391.4i 0 −29524.5 − 51137.9i 0
25.7 0 121.500 − 210.444i 0 4987.65 + 8638.86i 0 6340.58 + 44012.8i 0 −29524.5 − 51137.9i 0
37.1 0 121.500 + 210.444i 0 −4575.70 + 7925.35i 0 −7241.01 − 43873.6i 0 −29524.5 + 51137.9i 0
37.2 0 121.500 + 210.444i 0 −3945.47 + 6833.75i 0 39151.4 + 21082.9i 0 −29524.5 + 51137.9i 0
37.3 0 121.500 + 210.444i 0 −2471.06 + 4280.00i 0 −15790.8 + 41569.0i 0 −29524.5 + 51137.9i 0
37.4 0 121.500 + 210.444i 0 2050.38 − 3551.36i 0 −44391.9 − 2586.27i 0 −29524.5 + 51137.9i 0
37.5 0 121.500 + 210.444i 0 3193.56 − 5531.40i 0 42025.8 − 14531.2i 0 −29524.5 + 51137.9i 0
37.6 0 121.500 + 210.444i 0 4369.65 − 7568.45i 0 −2593.70 + 44391.4i 0 −29524.5 + 51137.9i 0
37.7 0 121.500 + 210.444i 0 4987.65 − 8638.86i 0 6340.58 − 44012.8i 0 −29524.5 + 51137.9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.7
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 84.12.i.a ✓ 14
3.b odd 2 1 252.12.k.b 14
7.c even 3 1 inner 84.12.i.a ✓ 14
21.h odd 6 1 252.12.k.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.i.a ✓ 14 1.a even 1 1 trivial
84.12.i.a ✓ 14 7.c even 3 1 inner
252.12.k.b 14 3.b odd 2 1
252.12.k.b 14 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 7218 T_{5}^{13} + 228016270 T_{5}^{12} - 1113050335212 T_{5}^{11} + \cdots + 66\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} - 243 T + 59049)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} - 7218 T^{13} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} - 35001 T^{13} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} - 54450 T^{13} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{7} - 767491 T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 1478880 T^{13} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{14} + 22875935 T^{13} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{14} - 62540568 T^{13} + \cdots + 91\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{7} - 51048864 T^{6} + \cdots + 49\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} - 188600405 T^{13} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{14} - 199685599 T^{13} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + 346934358 T^{6} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + 310350877 T^{6} + \cdots - 65\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} - 2771987346 T^{13} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} - 6487034184 T^{13} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{14} + 8183838888 T^{13} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} - 4069556330 T^{13} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{14} - 15766443531 T^{13} + \cdots + 94\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{7} + 16591642722 T^{6} + \cdots - 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 31685143839 T^{13} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} - 21999509987 T^{13} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{7} + 31526942994 T^{6} + \cdots - 46\!\cdots\!88)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} - 67041904680 T^{13} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} - 142041709050 T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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