Properties

Label 1386.2.c.a
Level $1386$
Weight $2$
Character orbit 1386.c
Analytic conductor $11.067$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 4 x^{10} - 8 x^{9} + 17 x^{8} - 16 x^{7} + 88 x^{6} - 48 x^{5} + 153 x^{4} - 216 x^{3} + 324 x^{2} + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( \beta_{1} - \beta_{5} ) q^{5} -\beta_{1} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( \beta_{1} - \beta_{5} ) q^{5} -\beta_{1} q^{7} - q^{8} + ( -\beta_{1} + \beta_{5} ) q^{10} + ( -\beta_{1} - \beta_{9} ) q^{11} + ( \beta_{7} - \beta_{11} ) q^{13} + \beta_{1} q^{14} + q^{16} + ( -1 - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{17} + ( -\beta_{1} + \beta_{3} + \beta_{5} - \beta_{11} ) q^{19} + ( \beta_{1} - \beta_{5} ) q^{20} + ( \beta_{1} + \beta_{9} ) q^{22} + ( 2 \beta_{3} - \beta_{7} - \beta_{11} ) q^{23} + ( -\beta_{2} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{25} + ( -\beta_{7} + \beta_{11} ) q^{26} -\beta_{1} q^{28} + ( 1 - \beta_{2} - 3 \beta_{6} - \beta_{8} + 2 \beta_{10} ) q^{29} + ( 1 - \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{31} - q^{32} + ( 1 + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{34} + ( 1 - \beta_{6} ) q^{35} + ( 2 - 2 \beta_{2} + 2 \beta_{8} ) q^{37} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{11} ) q^{38} + ( -\beta_{1} + \beta_{5} ) q^{40} + ( -3 + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{43} + ( -\beta_{1} - \beta_{9} ) q^{44} + ( -2 \beta_{3} + \beta_{7} + \beta_{11} ) q^{46} + ( -\beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{47} - q^{49} + ( \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{50} + ( \beta_{7} - \beta_{11} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{7} ) q^{53} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{55} + \beta_{1} q^{56} + ( -1 + \beta_{2} + 3 \beta_{6} + \beta_{8} - 2 \beta_{10} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{5} - \beta_{7} + \beta_{11} ) q^{61} + ( -1 + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{62} + q^{64} + ( -2 \beta_{2} + 2 \beta_{8} ) q^{65} + ( -3 - \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{67} + ( -1 - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{68} + ( -1 + \beta_{6} ) q^{70} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{11} ) q^{71} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{11} ) q^{73} + ( -2 + 2 \beta_{2} - 2 \beta_{8} ) q^{74} + ( -\beta_{1} + \beta_{3} + \beta_{5} - \beta_{11} ) q^{76} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{11} ) q^{77} + ( 5 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{79} + ( \beta_{1} - \beta_{5} ) q^{80} + ( 3 - \beta_{6} - 2 \beta_{8} + 2 \beta_{10} ) q^{82} + ( 1 - 4 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} ) q^{83} + ( 3 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{11} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{86} + ( \beta_{1} + \beta_{9} ) q^{88} + ( -4 \beta_{3} + \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{89} + ( 1 - \beta_{4} - \beta_{6} + \beta_{9} ) q^{91} + ( 2 \beta_{3} - \beta_{7} - \beta_{11} ) q^{92} + ( \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{94} + ( 3 - \beta_{2} - \beta_{6} + 3 \beta_{8} - 2 \beta_{10} ) q^{95} + ( 4 - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{4} - 12q^{8} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{4} - 12q^{8} + 4q^{11} + 12q^{16} - 16q^{17} - 4q^{22} - 4q^{25} - 16q^{29} - 12q^{32} + 16q^{34} + 8q^{35} + 24q^{37} - 16q^{41} + 4q^{44} - 12q^{49} + 4q^{50} - 8q^{55} + 16q^{58} + 12q^{64} - 48q^{67} - 16q^{68} - 8q^{70} - 24q^{74} - 8q^{77} + 16q^{82} - 16q^{83} - 4q^{88} + 48q^{95} + 48q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 4 x^{10} - 8 x^{9} + 17 x^{8} - 16 x^{7} + 88 x^{6} - 48 x^{5} + 153 x^{4} - 216 x^{3} + 324 x^{2} + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{11} - 97 \nu^{10} + 42 \nu^{9} - 301 \nu^{8} + 1004 \nu^{7} - 944 \nu^{6} + 1684 \nu^{5} - 4414 \nu^{4} + 3810 \nu^{3} - 3393 \nu^{2} + 13878 \nu + 567 \)\()/11502\)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{11} - 1767 \nu^{10} + 3122 \nu^{9} - 7417 \nu^{8} + 16600 \nu^{7} - 44516 \nu^{6} + 47456 \nu^{5} - 120456 \nu^{4} + 165492 \nu^{3} - 206631 \nu^{2} + 190674 \nu - 502281 \)\()/34506\)
\(\beta_{3}\)\(=\)\((\)\( -89 \nu^{11} - 206 \nu^{10} + 655 \nu^{9} - 1156 \nu^{8} + 2100 \nu^{7} - 9806 \nu^{6} + 6450 \nu^{5} - 22400 \nu^{4} + 32661 \nu^{3} - 46692 \nu^{2} + 21627 \nu - 145476 \)\()/11502\)
\(\beta_{4}\)\(=\)\((\)\( 335 \nu^{11} + 489 \nu^{10} - 2341 \nu^{9} - 535 \nu^{8} - 8648 \nu^{7} + 22222 \nu^{6} - 12382 \nu^{5} + 47184 \nu^{4} - 117567 \nu^{3} + 22599 \nu^{2} - 127413 \nu + 364257 \)\()/34506\)
\(\beta_{5}\)\(=\)\((\)\( -421 \nu^{11} - 495 \nu^{10} + 35 \nu^{9} - 97 \nu^{8} + 1168 \nu^{7} - 14324 \nu^{6} - 5602 \nu^{5} - 41190 \nu^{4} + 51597 \nu^{3} - 71307 \nu^{2} + 33291 \nu - 328293 \)\()/34506\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{11} - 12 \nu^{10} + 37 \nu^{9} - 50 \nu^{8} + 170 \nu^{7} - 172 \nu^{6} + 448 \nu^{5} - 456 \nu^{4} + 927 \nu^{3} - 648 \nu^{2} + 1863 \nu \)\()/486\)
\(\beta_{7}\)\(=\)\((\)\( -707 \nu^{11} - 510 \nu^{10} - 263 \nu^{9} + 808 \nu^{8} - 1486 \nu^{7} - 14530 \nu^{6} - 14636 \nu^{5} - 36216 \nu^{4} + 29169 \nu^{3} - 37800 \nu^{2} - 46737 \nu - 324648 \)\()/34506\)
\(\beta_{8}\)\(=\)\((\)\( -761 \nu^{11} - 489 \nu^{10} - 641 \nu^{9} + 109 \nu^{8} - 3706 \nu^{7} - 20518 \nu^{6} - 16160 \nu^{5} - 36960 \nu^{4} + 35775 \nu^{3} - 68607 \nu^{2} - 56619 \nu - 398763 \)\()/34506\)
\(\beta_{9}\)\(=\)\((\)\( -267 \nu^{11} + 1015 \nu^{10} - 1869 \nu^{9} + 4981 \nu^{8} - 10598 \nu^{7} + 19430 \nu^{6} - 29782 \nu^{5} + 61168 \nu^{4} - 72417 \nu^{3} + 94437 \nu^{2} - 126819 \nu + 155925 \)\()/11502\)
\(\beta_{10}\)\(=\)\((\)\( -1471 \nu^{11} + 2067 \nu^{10} - 3481 \nu^{9} + 12179 \nu^{8} - 24722 \nu^{7} + 18958 \nu^{6} - 77362 \nu^{5} + 99360 \nu^{4} - 76689 \nu^{3} + 161433 \nu^{2} - 332667 \nu - 157221 \)\()/34506\)
\(\beta_{11}\)\(=\)\((\)\( -305 \nu^{11} - 264 \nu^{10} + 208 \nu^{9} - 47 \nu^{8} + 533 \nu^{7} - 9547 \nu^{6} - 1634 \nu^{5} - 22962 \nu^{4} + 34377 \nu^{3} - 35091 \nu^{2} + 20439 \nu - 194643 \)\()/5751\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - 3 \beta_{9} + \beta_{8} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_{1} - 3\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + 9 \beta_{10} - 8 \beta_{9} - 5 \beta_{8} - 5 \beta_{7} + 5 \beta_{5} + 4 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} - 5 \beta_{1} + 12\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{11} + 2 \beta_{10} + \beta_{9} + 3 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 10 \beta_{1} - 12\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{11} - 13 \beta_{10} + 20 \beta_{9} - 5 \beta_{8} + 11 \beta_{7} - 6 \beta_{6} - 29 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 26 \beta_{2} - 7 \beta_{1} - 42\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(12 \beta_{11} + 23 \beta_{10} - 21 \beta_{9} - 49 \beta_{8} + 12 \beta_{7} + 3 \beta_{6} + 9 \beta_{5} + 26 \beta_{4} - 15 \beta_{3} + \beta_{2} - 15 \beta_{1} - 21\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-31 \beta_{11} + 24 \beta_{10} + 5 \beta_{9} - 10 \beta_{8} + 8 \beta_{7} + 18 \beta_{6} + 49 \beta_{5} + 5 \beta_{4} + 47 \beta_{3} - 2 \beta_{2} + 179 \beta_{1} - 78\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-16 \beta_{11} - 71 \beta_{10} + 89 \beta_{9} + 18 \beta_{8} + 44 \beta_{7} + 4 \beta_{5} - 42 \beta_{4} + 8 \beta_{3} + 60 \beta_{2} - 52 \beta_{1} + 9\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(61 \beta_{11} - 38 \beta_{10} - 35 \beta_{9} - 70 \beta_{8} + 112 \beta_{7} + 102 \beta_{6} - 67 \beta_{5} + 23 \beta_{4} + 91 \beta_{3} - 182 \beta_{2} - 281 \beta_{1} + 246\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(-156 \beta_{11} + 61 \beta_{10} - 123 \beta_{9} - 41 \beta_{8} - 108 \beta_{7} - 357 \beta_{6} + 159 \beta_{5} - 182 \beta_{4} + 375 \beta_{3} - 199 \beta_{2} + 423 \beta_{1} + 435\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(-359 \beta_{11} - 201 \beta_{10} - 26 \beta_{9} + 529 \beta_{8} - 209 \beta_{7} - 210 \beta_{6} + 767 \beta_{5} - 446 \beta_{4} - 332 \beta_{3} + 62 \beta_{2} - 1379 \beta_{1} + 210\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−1.26053 + 1.18788i
1.49380 0.876678i
−0.0383715 1.73163i
−0.748321 1.56205i
−0.833477 + 1.51833i
1.38690 1.03755i
1.38690 + 1.03755i
−0.833477 1.51833i
−0.748321 + 1.56205i
−0.0383715 + 1.73163i
1.49380 + 0.876678i
−1.26053 1.18788i
−1.00000 0 1.00000 3.46258i 0 1.00000i −1.00000 0 3.46258i
197.2 −1.00000 0 1.00000 3.35236i 0 1.00000i −1.00000 0 3.35236i
197.3 −1.00000 0 1.00000 2.50315i 0 1.00000i −1.00000 0 2.50315i
197.4 −1.00000 0 1.00000 1.15079i 0 1.00000i −1.00000 0 1.15079i
197.5 −1.00000 0 1.00000 0.968524i 0 1.00000i −1.00000 0 0.968524i
197.6 −1.00000 0 1.00000 0.494054i 0 1.00000i −1.00000 0 0.494054i
197.7 −1.00000 0 1.00000 0.494054i 0 1.00000i −1.00000 0 0.494054i
197.8 −1.00000 0 1.00000 0.968524i 0 1.00000i −1.00000 0 0.968524i
197.9 −1.00000 0 1.00000 1.15079i 0 1.00000i −1.00000 0 1.15079i
197.10 −1.00000 0 1.00000 2.50315i 0 1.00000i −1.00000 0 2.50315i
197.11 −1.00000 0 1.00000 3.35236i 0 1.00000i −1.00000 0 3.35236i
197.12 −1.00000 0 1.00000 3.46258i 0 1.00000i −1.00000 0 3.46258i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.c.a 12
3.b odd 2 1 1386.2.c.b yes 12
11.b odd 2 1 1386.2.c.b yes 12
33.d even 2 1 inner 1386.2.c.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.c.a 12 1.a even 1 1 trivial
1386.2.c.a 12 33.d even 2 1 inner
1386.2.c.b yes 12 3.b odd 2 1
1386.2.c.b yes 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{6} + 8 T_{17}^{5} - 20 T_{17}^{4} - 136 T_{17}^{3} + 306 T_{17}^{2} + 136 T_{17} - 376 \) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( T^{12} \)
$5$ \( 256 + 1600 T^{2} + 2628 T^{4} + 1600 T^{6} + 356 T^{8} + 32 T^{10} + T^{12} \)
$7$ \( ( 1 + T^{2} )^{6} \)
$11$ \( 1771561 - 644204 T + 87846 T^{2} - 37268 T^{3} - 6897 T^{4} + 5192 T^{5} - 556 T^{6} + 472 T^{7} - 57 T^{8} - 28 T^{9} + 6 T^{10} - 4 T^{11} + T^{12} \)
$13$ \( 1024 + 51200 T^{2} + 63872 T^{4} + 19776 T^{6} + 2212 T^{8} + 84 T^{10} + T^{12} \)
$17$ \( ( -376 + 136 T + 306 T^{2} - 136 T^{3} - 20 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$19$ \( 16 + 528 T^{2} + 1876 T^{4} + 2160 T^{6} + 872 T^{8} + 68 T^{10} + T^{12} \)
$23$ \( 79709184 + 31684608 T^{2} + 4680064 T^{4} + 325696 T^{6} + 11236 T^{8} + 180 T^{10} + T^{12} \)
$29$ \( ( -512 + 2816 T + 708 T^{2} - 624 T^{3} - 84 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$31$ \( ( 3656 - 4296 T + 1106 T^{2} + 176 T^{3} - 76 T^{4} + T^{6} )^{2} \)
$37$ \( ( 10688 - 6976 T - 304 T^{2} + 608 T^{3} - 36 T^{4} - 12 T^{5} + T^{6} )^{2} \)
$41$ \( ( -56 + 552 T - 894 T^{2} - 624 T^{3} - 76 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$43$ \( 541696 + 2596864 T^{2} + 2974080 T^{4} + 358336 T^{6} + 14564 T^{8} + 212 T^{10} + T^{12} \)
$47$ \( 7322567184 + 1265350032 T^{2} + 84078868 T^{4} + 2732144 T^{6} + 45256 T^{8} + 356 T^{10} + T^{12} \)
$53$ \( 1048576 + 109248512 T^{2} + 24544256 T^{4} + 1709056 T^{6} + 40128 T^{8} + 352 T^{10} + T^{12} \)
$59$ \( 339738624 + 191102976 T^{2} + 32837632 T^{4} + 1861632 T^{6} + 41344 T^{8} + 352 T^{10} + T^{12} \)
$61$ \( 183223296 + 67295232 T^{2} + 8829568 T^{4} + 540096 T^{6} + 16036 T^{8} + 212 T^{10} + T^{12} \)
$67$ \( ( -42752 - 41792 T - 12860 T^{2} - 1136 T^{3} + 116 T^{4} + 24 T^{5} + T^{6} )^{2} \)
$71$ \( 16516096 + 37480448 T^{2} + 7635584 T^{4} + 566464 T^{6} + 18276 T^{8} + 244 T^{10} + T^{12} \)
$73$ \( 14868736 + 522608192 T^{2} + 120308036 T^{4} + 5620768 T^{6} + 84324 T^{8} + 496 T^{10} + T^{12} \)
$79$ \( 14992384 + 15890688 T^{2} + 5790480 T^{4} + 838368 T^{6} + 42776 T^{8} + 424 T^{10} + T^{12} \)
$83$ \( ( 94856 + 45240 T - 750 T^{2} - 2584 T^{3} - 260 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$89$ \( 133476238336 + 32154304512 T^{2} + 1540955776 T^{4} + 28416576 T^{6} + 225380 T^{8} + 788 T^{10} + T^{12} \)
$97$ \( ( -112896 - 67200 T - 2960 T^{2} + 2560 T^{3} - 8 T^{4} - 24 T^{5} + T^{6} )^{2} \)
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