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Properties

Label 84.9.m.b
Level $84$
Weight $9$
Character orbit 84.m
Analytic conductor $34.220$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{10}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 27 \beta_1 + 54) q^{3} + ( - \beta_{3} + 16 \beta_1 + 16) q^{5} + (\beta_{5} + \beta_{3} - 196 \beta_1 + 114) q^{7} + ( - 2187 \beta_1 + 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 27 \beta_1 + 54) q^{3} + ( - \beta_{3} + 16 \beta_1 + 16) q^{5} + (\beta_{5} + \beta_{3} - 196 \beta_1 + 114) q^{7} + ( - 2187 \beta_1 + 2187) q^{9} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 2987 \beta_1) q^{11} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots + 1915) q^{13}+ \cdots + (2187 \beta_{7} + 2187 \beta_{5} + 2187 \beta_{4} + \cdots - 6532569) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + \cdots + 45\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!43 \nu^{11} + \cdots - 14\!\cdots\!20 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34\!\cdots\!83 \nu^{11} + \cdots + 41\!\cdots\!04 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 42\!\cdots\!13 \nu^{11} + \cdots + 17\!\cdots\!08 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\!\cdots\!23 \nu^{11} + \cdots - 86\!\cdots\!88 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 92\!\cdots\!02 \nu^{11} + \cdots + 10\!\cdots\!68 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!68 \nu^{11} + \cdots + 78\!\cdots\!92 ) / 28\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!39 \nu^{11} + \cdots - 44\!\cdots\!40 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!45 \nu^{11} + \cdots - 20\!\cdots\!48 ) / 50\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!29 \nu^{11} + \cdots - 79\!\cdots\!00 ) / 50\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!71 \nu^{11} + \cdots + 25\!\cdots\!72 ) / 20\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!43 \nu^{11} + \cdots - 99\!\cdots\!36 ) / 20\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + 2\beta_{8} + 11\beta_{5} - 2\beta_{4} - 13\beta_{3} + 5\beta_{2} + 89\beta _1 - 4 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 49 \beta_{11} - 98 \beta_{10} + 379 \beta_{9} + \beta_{8} - 357 \beta_{6} + 460 \beta_{5} - 3002 \beta_{4} - 3851 \beta_{3} + 6992 \beta_{2} + 8294560 \beta _1 - 8295551 ) / 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35084 \beta_{11} - 35084 \beta_{10} + 789666 \beta_{9} - 376181 \beta_{8} - 275352 \beta_{7} - 5422393 \beta_{5} - 4661151 \beta_{4} + 5080849 \beta_{3} + 6316864 \beta_{2} + \cdots - 5863038373 ) / 504 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 53732714 \beta_{11} + 26866357 \beta_{10} + 42409788 \beta_{9} - 238180922 \beta_{8} - 173747049 \beta_{7} + 173747049 \beta_{6} - 2066402749 \beta_{5} + \cdots + 602856833 ) / 504 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8924207908 \beta_{11} + 17848415816 \beta_{10} - 91153716975 \beta_{9} - 55734876919 \beta_{8} + 65807686308 \beta_{6} + 94177321174 \beta_{5} + \cdots + 13\!\cdots\!03 ) / 504 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4764138914563 \beta_{11} + 4764138914563 \beta_{10} - 56310552509697 \beta_{9} + 33032234588515 \beta_{8} + 31905883209375 \beta_{7} + \cdots + 66\!\cdots\!34 ) / 504 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12\!\cdots\!96 \beta_{11} - 637399972567148 \beta_{10} + \cdots - 18\!\cdots\!84 ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 29\!\cdots\!67 \beta_{11} + \cdots - 42\!\cdots\!61 ) / 168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 38\!\cdots\!56 \beta_{11} + \cdots - 56\!\cdots\!67 ) / 504 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11\!\cdots\!10 \beta_{11} + \cdots + 16\!\cdots\!09 ) / 168 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 76\!\cdots\!72 \beta_{11} + \cdots + 11\!\cdots\!37 ) / 504 \) Copy content Toggle raw display

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

Character values

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

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

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
44.6586 + 77.3509i
−72.3408 − 125.298i
−28.8366 − 49.9465i
−122.377 − 211.963i
221.993 + 384.503i
−41.5970 − 72.0480i
44.6586 − 77.3509i
−72.3408 + 125.298i
−28.8366 + 49.9465i
−122.377 + 211.963i
221.993 − 384.503i
−41.5970 + 72.0480i
0 40.5000 − 23.3827i 0 −939.615 − 542.487i 0 1451.57 + 1912.52i 0 1093.50 − 1894.00i 0
61.2 0 40.5000 − 23.3827i 0 −225.043 − 129.928i 0 597.275 − 2325.52i 0 1093.50 − 1894.00i 0
61.3 0 40.5000 − 23.3827i 0 −203.366 − 117.413i 0 1130.34 − 2118.29i 0 1093.50 − 1894.00i 0
61.4 0 40.5000 − 23.3827i 0 −133.903 − 77.3091i 0 −2234.88 + 877.558i 0 1093.50 − 1894.00i 0
61.5 0 40.5000 − 23.3827i 0 632.551 + 365.204i 0 984.437 + 2189.91i 0 1093.50 − 1894.00i 0
61.6 0 40.5000 − 23.3827i 0 1011.88 + 584.207i 0 −1829.74 − 1554.62i 0 1093.50 − 1894.00i 0
73.1 0 40.5000 + 23.3827i 0 −939.615 + 542.487i 0 1451.57 − 1912.52i 0 1093.50 + 1894.00i 0
73.2 0 40.5000 + 23.3827i 0 −225.043 + 129.928i 0 597.275 + 2325.52i 0 1093.50 + 1894.00i 0
73.3 0 40.5000 + 23.3827i 0 −203.366 + 117.413i 0 1130.34 + 2118.29i 0 1093.50 + 1894.00i 0
73.4 0 40.5000 + 23.3827i 0 −133.903 + 77.3091i 0 −2234.88 − 877.558i 0 1093.50 + 1894.00i 0
73.5 0 40.5000 + 23.3827i 0 632.551 − 365.204i 0 984.437 − 2189.91i 0 1093.50 + 1894.00i 0
73.6 0 40.5000 + 23.3827i 0 1011.88 − 584.207i 0 −1829.74 + 1554.62i 0 1093.50 + 1894.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.6
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.b ✓ 12
3.b odd 2 1 252.9.z.d 12
7.d odd 6 1 inner 84.9.m.b ✓ 12
21.g even 6 1 252.9.z.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.m.b ✓ 12 1.a even 1 1 trivial
84.9.m.b ✓ 12 7.d odd 6 1 inner
252.9.z.d 12 3.b odd 2 1
252.9.z.d 12 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 285 T_{5}^{11} - 1570602 T_{5}^{10} + 455337945 T_{5}^{9} + 2118409775334 T_{5}^{8} - 311260236586983 T_{5}^{7} + \cdots + 76\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 81 T + 2187)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} - 285 T^{11} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} - 198 T^{11} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + 17919 T^{11} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + 5107913481 T^{10} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + 205782 T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} - 74313 T^{11} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{12} + 62832 T^{11} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{6} + 287727 T^{5} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 1442952 T^{11} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} + 2058621 T^{11} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{12} + 44050323101928 T^{10} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3860661 T^{5} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 12088194 T^{11} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + 5506743 T^{11} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} - 7511901 T^{11} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + 37215576 T^{11} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + 36824553 T^{11} + \cdots + 65\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + 15005778 T^{5} + \cdots - 52\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 95080185 T^{11} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} - 8514456 T^{11} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 78\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} - 83038554 T^{11} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
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