Properties

Label 84.9.m.a
Level $84$
Weight $9$
Character orbit 84.m
Analytic conductor $34.220$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.2198032451\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + (27 \beta_1 - 54) q^{3} + (\beta_{5} + 93 \beta_1 + 92) q^{5} + (\beta_{9} - \beta_{4} + \beta_{3} - 35 \beta_1 + 139) q^{7} + ( - 2187 \beta_1 + 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (27 \beta_1 - 54) q^{3} + (\beta_{5} + 93 \beta_1 + 92) q^{5} + (\beta_{9} - \beta_{4} + \beta_{3} - 35 \beta_1 + 139) q^{7} + ( - 2187 \beta_1 + 2187) q^{9} + (2 \beta_{9} + \beta_{8} + \beta_{7} + 4 \beta_{6} + 10 \beta_{5} - 3 \beta_{3} - 5 \beta_{2} + \cdots - 6) q^{11}+ \cdots + (4374 \beta_{9} + 2187 \beta_{8} + 2187 \beta_{6} + 10935 \beta_{5} + \cdots - 395847) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 405 q^{3} + 1389 q^{5} + 1217 q^{7} + 10935 q^{9} - 879 q^{11} - 75006 q^{15} - 13674 q^{17} - 29268 q^{19} - 42363 q^{21} + 312732 q^{23} - 22052 q^{25} - 289794 q^{29} + 242787 q^{31} + 71199 q^{33} + 1209372 q^{35} + 1913308 q^{37} - 1232334 q^{39} - 861848 q^{43} + 3037743 q^{45} - 305448 q^{47} + 9821659 q^{49} + 369198 q^{51} - 10663233 q^{53} + 1580472 q^{57} + 18410871 q^{59} - 13937808 q^{61} + 769824 q^{63} - 14966808 q^{65} - 20722822 q^{67} + 113032584 q^{71} + 43436322 q^{73} + 1786212 q^{75} - 98823405 q^{77} - 42189637 q^{79} - 23914845 q^{81} + 142602108 q^{85} + 11736657 q^{87} + 67171914 q^{89} - 246091266 q^{91} - 6555249 q^{93} - 140649894 q^{95} - 3844746 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} - 71600654137860 x^{3} + 119846615988780 x^{2} + \cdots + 15\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\!\cdots\!39 \nu^{9} + \cdots + 25\!\cdots\!80 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!52 \nu^{9} + \cdots - 10\!\cdots\!10 ) / 57\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!70 \nu^{9} + \cdots - 25\!\cdots\!70 ) / 19\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!33 \nu^{9} + \cdots + 33\!\cdots\!20 ) / 24\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!17 \nu^{9} + \cdots - 42\!\cdots\!50 ) / 57\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!39 \nu^{9} + \cdots - 14\!\cdots\!80 ) / 76\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\!\cdots\!97 \nu^{9} + \cdots - 16\!\cdots\!40 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!98 \nu^{9} + \cdots - 49\!\cdots\!50 ) / 19\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!87 \nu^{9} + \cdots - 30\!\cdots\!00 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -7\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 7\beta_{4} - \beta_{3} - 18\beta_{2} + 10\beta _1 - 2 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 133 \beta_{9} - 25 \beta_{7} + 641 \beta_{6} - 612 \beta_{5} + 308 \beta_{4} + 154 \beta_{3} + 1224 \beta_{2} - 1286234 \beta _1 + 1285468 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 98350 \beta_{9} + 9962 \beta_{8} + 19924 \beta_{7} - 44872 \beta_{6} + 204264 \beta_{5} + 40859 \beta_{4} + 18278 \beta_{3} - 204264 \beta_{2} + 92222164 \beta _1 - 46106924 ) / 63 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7241731 \beta_{9} - 1087531 \beta_{8} - 1087531 \beta_{7} + 15842271 \beta_{6} - 31426884 \beta_{5} - 1339835 \beta_{4} - 12891323 \beta_{3} + 15713442 \beta_{2} + \cdots + 12762494 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2674551586 \beta_{9} + 817640630 \beta_{8} + 408820315 \beta_{7} - 1888573513 \beta_{6} + 9578395467 \beta_{5} + 3869349974 \beta_{4} + \cdots + 2924353887244 ) / 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17306620562 \beta_{9} - 4718798746 \beta_{8} + 8653310281 \beta_{6} - 55189236609 \beta_{5} - 28925053171 \beta_{4} - 47985350151 \beta_{3} + \cdots - 77852302903406 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 38202992333786 \beta_{9} + 17818961417759 \beta_{8} - 17818961417759 \beta_{7} + 62423963401070 \beta_{6} + 127412996300408 \beta_{4} + \cdots + 31\!\cdots\!16 ) / 63 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 51\!\cdots\!84 \beta_{9} + \cdots - 24\!\cdots\!06 ) / 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 74\!\cdots\!70 \beta_{9} + \cdots + 81\!\cdots\!52 ) / 63 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−3.70642 + 2.13991i
38.0902 21.9914i
139.521 80.5522i
−190.225 + 109.826i
16.3207 9.42278i
−3.70642 2.13991i
38.0902 + 21.9914i
139.521 + 80.5522i
−190.225 109.826i
16.3207 + 9.42278i
0 −40.5000 + 23.3827i 0 −489.814 282.795i 0 −1456.69 1908.63i 0 1093.50 1894.00i 0
61.2 0 −40.5000 + 23.3827i 0 −374.307 216.106i 0 83.7630 + 2399.54i 0 1093.50 1894.00i 0
61.3 0 −40.5000 + 23.3827i 0 111.592 + 64.4274i 0 2392.99 + 195.963i 0 1093.50 1894.00i 0
61.4 0 −40.5000 + 23.3827i 0 656.878 + 379.249i 0 1906.96 1458.87i 0 1093.50 1894.00i 0
61.5 0 −40.5000 + 23.3827i 0 790.151 + 456.194i 0 −2318.52 + 623.902i 0 1093.50 1894.00i 0
73.1 0 −40.5000 23.3827i 0 −489.814 + 282.795i 0 −1456.69 + 1908.63i 0 1093.50 + 1894.00i 0
73.2 0 −40.5000 23.3827i 0 −374.307 + 216.106i 0 83.7630 2399.54i 0 1093.50 + 1894.00i 0
73.3 0 −40.5000 23.3827i 0 111.592 64.4274i 0 2392.99 195.963i 0 1093.50 + 1894.00i 0
73.4 0 −40.5000 23.3827i 0 656.878 379.249i 0 1906.96 + 1458.87i 0 1093.50 + 1894.00i 0
73.5 0 −40.5000 23.3827i 0 790.151 456.194i 0 −2318.52 623.902i 0 1093.50 + 1894.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.m.a 10
3.b odd 2 1 252.9.z.b 10
7.d odd 6 1 inner 84.9.m.a 10
21.g even 6 1 252.9.z.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.m.a 10 1.a even 1 1 trivial
84.9.m.a 10 7.d odd 6 1 inner
252.9.z.b 10 3.b odd 2 1
252.9.z.b 10 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 1389 T_{5}^{9} - 876 T_{5}^{8} + 894492387 T_{5}^{7} - 19821737736 T_{5}^{6} - 381886886416725 T_{5}^{5} + \cdots + 47\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81 T + 2187)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 1389 T^{9} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} - 1217 T^{9} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{10} + 879 T^{9} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + 4745027382 T^{8} + \cdots + 67\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{10} + 13674 T^{9} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + 29268 T^{9} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( T^{10} - 312732 T^{9} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{5} + 144897 T^{4} + \cdots + 37\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 242787 T^{9} + \cdots + 46\!\cdots\!43 \) Copy content Toggle raw display
$37$ \( T^{10} - 1913308 T^{9} + \cdots + 82\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{10} + 44132018588988 T^{8} + \cdots + 31\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{5} + 430924 T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 305448 T^{9} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + 10663233 T^{9} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{10} - 18410871 T^{9} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{10} + 13937808 T^{9} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + 20722822 T^{9} + \cdots + 62\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{5} - 56516292 T^{4} + \cdots - 22\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 43436322 T^{9} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + 42189637 T^{9} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 34\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{10} - 67171914 T^{9} + \cdots + 26\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
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