Properties

Label 162.11.d.d
Level $162$
Weight $11$
Character orbit 162.d
Analytic conductor $102.928$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.927874933\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17318914560000.97
Defining polynomial: \(x^{8} - 4 x^{7} - 82 x^{6} + 260 x^{5} + 2477 x^{4} - 5392 x^{3} - 31616 x^{2} + 34356 x + 161859\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} -512 \beta_{1} q^{4} + ( 26 \beta_{3} + \beta_{4} + 26 \beta_{5} ) q^{5} + ( 11278 + 11278 \beta_{1} - \beta_{7} ) q^{7} + 512 \beta_{5} q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} -512 \beta_{1} q^{4} + ( 26 \beta_{3} + \beta_{4} + 26 \beta_{5} ) q^{5} + ( 11278 + 11278 \beta_{1} - \beta_{7} ) q^{7} + 512 \beta_{5} q^{8} + ( -13440 - 8 \beta_{6} ) q^{10} + ( 35 \beta_{2} + 3766 \beta_{3} ) q^{11} + ( 68810 \beta_{1} - 20 \beta_{6} + 20 \beta_{7} ) q^{13} + ( -11294 \beta_{3} + 64 \beta_{4} - 11294 \beta_{5} ) q^{14} + ( -262144 - 262144 \beta_{1} ) q^{16} + ( 236 \beta_{2} - 236 \beta_{4} + 73832 \beta_{5} ) q^{17} + ( -392182 - 97 \beta_{6} ) q^{19} + ( 512 \beta_{2} + 13312 \beta_{3} ) q^{20} + ( 1932672 \beta_{1} - 280 \beta_{6} + 280 \beta_{7} ) q^{22} + ( -91988 \beta_{3} - 826 \beta_{4} - 91988 \beta_{5} ) q^{23} + ( 8433095 + 8433095 \beta_{1} + 420 \beta_{7} ) q^{25} + ( 1280 \beta_{2} - 1280 \beta_{4} - 68490 \beta_{5} ) q^{26} + ( 5774336 - 512 \beta_{6} ) q^{28} + ( -2947 \beta_{2} - 763022 \beta_{3} ) q^{29} + ( -5446462 \beta_{1} - 3045 \beta_{6} + 3045 \beta_{7} ) q^{31} + ( 262144 \beta_{3} + 262144 \beta_{5} ) q^{32} + ( -37771776 - 37771776 \beta_{1} + 1888 \beta_{7} ) q^{34} + ( -9598 \beta_{2} + 9598 \beta_{4} - 1937092 \beta_{5} ) q^{35} + ( -17753542 - 5684 \beta_{6} ) q^{37} + ( 6208 \beta_{2} + 390630 \beta_{3} ) q^{38} + ( 6881280 \beta_{1} - 4096 \beta_{6} + 4096 \beta_{7} ) q^{40} + ( -4094500 \beta_{3} + 13414 \beta_{4} - 4094500 \beta_{5} ) q^{41} + ( 117672166 + 117672166 \beta_{1} - 6783 \beta_{7} ) q^{43} + ( 17920 \beta_{2} - 17920 \beta_{4} - 1928192 \beta_{5} ) q^{44} + ( 47203584 + 6608 \beta_{6} ) q^{46} + ( -14460 \beta_{2} + 4156680 \beta_{3} ) q^{47} + ( -12514605 \beta_{1} + 22556 \beta_{6} - 22556 \beta_{7} ) q^{49} + ( -8426375 \beta_{3} - 26880 \beta_{4} - 8426375 \beta_{5} ) q^{50} + ( 35230720 + 35230720 \beta_{1} + 10240 \beta_{7} ) q^{52} + ( -59339 \beta_{2} + 59339 \beta_{4} + 9200542 \beta_{5} ) q^{53} + ( 675339840 + 37548 \beta_{6} ) q^{55} + ( 32768 \beta_{2} - 5782528 \beta_{3} ) q^{56} + ( -391044480 \beta_{1} + 23576 \beta_{6} - 23576 \beta_{7} ) q^{58} + ( 17244994 \beta_{3} + 23273 \beta_{4} + 17244994 \beta_{5} ) q^{59} + ( 296009686 + 296009686 \beta_{1} - 19644 \beta_{7} ) q^{61} + ( 194880 \beta_{2} - 194880 \beta_{4} + 5495182 \beta_{5} ) q^{62} -134217728 q^{64} + ( -102410 \beta_{2} - 46395460 \beta_{3} ) q^{65} + ( -74341462 \beta_{1} + 49917 \beta_{6} - 49917 \beta_{7} ) q^{67} + ( 37801984 \beta_{3} - 120832 \beta_{4} + 37801984 \beta_{5} ) q^{68} + ( 990562560 + 990562560 \beta_{1} - 76784 \beta_{7} ) q^{70} + ( -213598 \beta_{2} + 213598 \beta_{4} + 14573468 \beta_{5} ) q^{71} + ( 1633567250 - 46224 \beta_{6} ) q^{73} + ( 363776 \beta_{2} + 17662598 \beta_{3} ) q^{74} + ( 200797184 \beta_{1} - 49664 \beta_{6} + 49664 \beta_{7} ) q^{76} + ( -35542556 \beta_{3} + 153146 \beta_{4} - 35542556 \beta_{5} ) q^{77} + ( -49820642 - 49820642 \beta_{1} + 160363 \beta_{7} ) q^{79} + ( 262144 \beta_{2} - 262144 \beta_{4} - 6815744 \beta_{5} ) q^{80} + ( 2094667008 - 107312 \beta_{6} ) q^{82} + ( -388359 \beta_{2} + 24181890 \beta_{3} ) q^{83} + ( -3220128000 \beta_{1} - 540624 \beta_{6} + 540624 \beta_{7} ) q^{85} + ( -117780694 \beta_{3} + 434112 \beta_{4} - 117780694 \beta_{5} ) q^{86} + ( 989528064 + 989528064 \beta_{1} + 143360 \beta_{7} ) q^{88} + ( -587062 \beta_{2} + 587062 \beta_{4} + 120006236 \beta_{5} ) q^{89} + ( 2079308020 - 156750 \beta_{6} ) q^{91} + ( -422912 \beta_{2} - 47097856 \beta_{3} ) q^{92} + ( 2126369280 \beta_{1} + 115680 \beta_{6} - 115680 \beta_{7} ) q^{94} + ( -226537772 \beta_{3} - 555142 \beta_{4} - 226537772 \beta_{5} ) q^{95} + ( 9794088766 + 9794088766 \beta_{1} + 79756 \beta_{7} ) q^{97} + ( -1443584 \beta_{2} + 1443584 \beta_{4} + 12153709 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2048 q^{4} + 45112 q^{7} + O(q^{10}) \) \( 8 q + 2048 q^{4} + 45112 q^{7} - 107520 q^{10} - 275240 q^{13} - 1048576 q^{16} - 3137456 q^{19} - 7730688 q^{22} + 33732380 q^{25} + 46194688 q^{28} + 21785848 q^{31} - 151087104 q^{34} - 142028336 q^{37} - 27525120 q^{40} + 470688664 q^{43} + 377628672 q^{46} + 50058420 q^{49} + 140922880 q^{52} + 5402718720 q^{55} + 1564177920 q^{58} + 1184038744 q^{61} - 1073741824 q^{64} + 297365848 q^{67} + 3962250240 q^{70} + 13068538000 q^{73} - 803188736 q^{76} - 199282568 q^{79} + 16757336064 q^{82} + 12880512000 q^{85} + 3958112256 q^{88} + 16634464160 q^{91} - 8505477120 q^{94} + 39176355064 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 82 x^{6} + 260 x^{5} + 2477 x^{4} - 5392 x^{3} - 31616 x^{2} + 34356 x + 161859\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} - 6 \nu^{5} - 211 \nu^{4} + 432 \nu^{3} + 6275 \nu^{2} - 6492 \nu - 57465 \)\()/13746\)
\(\beta_{2}\)\(=\)\((\)\( -6736 \nu^{7} - 4259056 \nu^{6} + 13438472 \nu^{5} + 450282296 \nu^{4} - 945290272 \nu^{3} - 28122112960 \nu^{2} + 28839704256 \nu + 431997408168 \)\()/45423657\)
\(\beta_{3}\)\(=\)\((\)\( -26944 \nu^{7} + 94304 \nu^{6} + 2362304 \nu^{5} - 6141520 \nu^{4} - 80967040 \nu^{3} + 127639232 \nu^{2} + 884063664 \nu - 463512000 \)\()/45423657\)
\(\beta_{4}\)\(=\)\((\)\(25180 \nu^{7} + 88776484 \nu^{6} - 268154120 \nu^{5} - 5691779540 \nu^{4} + 11865395680 \nu^{3} + 113195424340 \nu^{2} - 119327084172 \nu - 606875544888\)\()/45423657\)
\(\beta_{5}\)\(=\)\((\)\( 1616 \nu^{7} - 5656 \nu^{6} - 108904 \nu^{5} + 286400 \nu^{4} + 2507000 \nu^{3} - 4049728 \nu^{2} - 17321976 \nu + 9345624 \)\()/574983\)
\(\beta_{6}\)\(=\)\((\)\( -484992 \nu^{7} + 1697472 \nu^{6} + 42521472 \nu^{5} - 110547360 \nu^{4} - 1457406720 \nu^{3} + 2297506176 \nu^{2} + 28995159168 \nu - 14884222608 \)\()/5047073\)
\(\beta_{7}\)\(=\)\((\)\( 3716352 \nu^{7} - 13007232 \nu^{6} - 310292784 \nu^{5} + 808250040 \nu^{4} + 7873873344 \nu^{3} - 12625563672 \nu^{2} - 47236696128 \nu + 25749860040 \)\()/5047073\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - 162 \beta_{3} + 1296\)\()/2592\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 160 \beta_{3} - 8 \beta_{2} - 5184 \beta_{1} + 55728\)\()/2592\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 14 \beta_{6} + 162 \beta_{5} - 5223 \beta_{3} - 6 \beta_{2} - 3888 \beta_{1} + 41472\)\()/1296\)
\(\nu^{4}\)\(=\)\((\)\(-12 \beta_{7} + 55 \beta_{6} + 640 \beta_{5} + 32 \beta_{4} - 20648 \beta_{3} - 384 \beta_{2} - 679104 \beta_{1} + 1242864\)\()/2592\)
\(\nu^{5}\)\(=\)\((\)\(-460 \beta_{7} + 945 \beta_{6} + 70288 \beta_{5} + 80 \beta_{4} - 390772 \beta_{3} - 940 \beta_{2} - 1684800 \beta_{1} + 2969136\)\()/2592\)
\(\nu^{6}\)\(=\)\((\)\(-675 \beta_{7} + 1349 \beta_{6} + 104200 \beta_{5} + 1808 \beta_{4} - 559098 \beta_{3} - 7820 \beta_{2} - 20470320 \beta_{1} + 12974256\)\()/1296\)
\(\nu^{7}\)\(=\)\((\)\(-24290 \beta_{7} + 33167 \beta_{6} + 5772368 \beta_{5} + 12376 \beta_{4} - 12520766 \beta_{3} - 51464 \beta_{2} - 137404512 \beta_{1} + 80524368\)\()/2592\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−5.33452 + 0.707107i
3.88503 + 0.707107i
6.33452 0.707107i
−2.88503 0.707107i
−5.33452 0.707107i
3.88503 0.707107i
6.33452 + 0.707107i
−2.88503 + 0.707107i
−19.5959 + 11.3137i 0 256.000 443.405i −3144.08 1815.24i 0 11613.3 + 20114.8i 11585.2i 0 82148.2
53.2 −19.5959 + 11.3137i 0 256.000 443.405i 4172.87 + 2409.21i 0 −335.265 580.696i 11585.2i 0 −109028.
53.3 19.5959 11.3137i 0 256.000 443.405i −4172.87 2409.21i 0 −335.265 580.696i 11585.2i 0 −109028.
53.4 19.5959 11.3137i 0 256.000 443.405i 3144.08 + 1815.24i 0 11613.3 + 20114.8i 11585.2i 0 82148.2
107.1 −19.5959 11.3137i 0 256.000 + 443.405i −3144.08 + 1815.24i 0 11613.3 20114.8i 11585.2i 0 82148.2
107.2 −19.5959 11.3137i 0 256.000 + 443.405i 4172.87 2409.21i 0 −335.265 + 580.696i 11585.2i 0 −109028.
107.3 19.5959 + 11.3137i 0 256.000 + 443.405i −4172.87 + 2409.21i 0 −335.265 + 580.696i 11585.2i 0 −109028.
107.4 19.5959 + 11.3137i 0 256.000 + 443.405i 3144.08 1815.24i 0 11613.3 20114.8i 11585.2i 0 82148.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.11.d.d 8
3.b odd 2 1 inner 162.11.d.d 8
9.c even 3 1 6.11.b.a 4
9.c even 3 1 inner 162.11.d.d 8
9.d odd 6 1 6.11.b.a 4
9.d odd 6 1 inner 162.11.d.d 8
36.f odd 6 1 48.11.e.d 4
36.h even 6 1 48.11.e.d 4
45.h odd 6 1 150.11.d.a 4
45.j even 6 1 150.11.d.a 4
45.k odd 12 2 150.11.b.a 8
45.l even 12 2 150.11.b.a 8
72.j odd 6 1 192.11.e.g 4
72.l even 6 1 192.11.e.h 4
72.n even 6 1 192.11.e.g 4
72.p odd 6 1 192.11.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 9.c even 3 1
6.11.b.a 4 9.d odd 6 1
48.11.e.d 4 36.f odd 6 1
48.11.e.d 4 36.h even 6 1
150.11.b.a 8 45.k odd 12 2
150.11.b.a 8 45.l even 12 2
150.11.d.a 4 45.h odd 6 1
150.11.d.a 4 45.j even 6 1
162.11.d.d 8 1.a even 1 1 trivial
162.11.d.d 8 3.b odd 2 1 inner
162.11.d.d 8 9.c even 3 1 inner
162.11.d.d 8 9.d odd 6 1 inner
192.11.e.g 4 72.j odd 6 1
192.11.e.g 4 72.n even 6 1
192.11.e.h 4 72.l even 6 1
192.11.e.h 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 36397440 T_{5}^{6} + \)\(10\!\cdots\!00\)\( T_{5}^{4} - \)\(11\!\cdots\!00\)\( T_{5}^{2} + \)\(93\!\cdots\!00\)\( \) acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 262144 - 512 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( \)\(93\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( T^{2} + 1018764391219200 T^{4} - 36397440 T^{6} + T^{8} \)
$7$ \( ( 242551843253776 + 351288858256 T + 524347212 T^{2} - 22556 T^{3} + T^{4} )^{2} \)
$11$ \( \)\(45\!\cdots\!76\)\( - \)\(12\!\cdots\!36\)\( T^{2} + \)\(31\!\cdots\!72\)\( T^{4} - 58313211264 T^{6} + T^{8} \)
$13$ \( ( \)\(27\!\cdots\!00\)\( - 7207452241598000 T + 71311392300 T^{2} + 137620 T^{3} + T^{4} )^{2} \)
$17$ \( ( \)\(32\!\cdots\!84\)\( + 7560967182336 T^{2} + T^{4} )^{2} \)
$19$ \( ( -1189491369116 + 784364 T + T^{2} )^{4} \)
$23$ \( \)\(37\!\cdots\!76\)\( - \)\(20\!\cdots\!84\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{4} - 33055507478016 T^{6} + T^{8} \)
$29$ \( \)\(42\!\cdots\!00\)\( - \)\(18\!\cdots\!00\)\( T^{2} + \)\(80\!\cdots\!00\)\( T^{4} - 907304099736960 T^{6} + T^{8} \)
$31$ \( ( \)\(16\!\cdots\!36\)\( + \)\(14\!\cdots\!44\)\( T + 1412734376056332 T^{2} - 10892924 T^{3} + T^{4} )^{2} \)
$37$ \( ( -4297319054834396 + 35507084 T + T^{2} )^{4} \)
$41$ \( \)\(82\!\cdots\!16\)\( - \)\(67\!\cdots\!36\)\( T^{2} + \)\(52\!\cdots\!52\)\( T^{4} - 23561404561257984 T^{6} + T^{8} \)
$43$ \( ( \)\(52\!\cdots\!56\)\( - \)\(17\!\cdots\!12\)\( T + 48108812125929708 T^{2} - 235344332 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(67\!\cdots\!00\)\( - \)\(65\!\cdots\!00\)\( T^{2} + \)\(60\!\cdots\!00\)\( T^{4} - 25124763600230400 T^{6} + T^{8} \)
$53$ \( ( \)\(37\!\cdots\!04\)\( + 212636466457531776 T^{2} + T^{4} )^{2} \)
$59$ \( \)\(41\!\cdots\!16\)\( - \)\(65\!\cdots\!96\)\( T^{2} + \)\(84\!\cdots\!72\)\( T^{4} - 324064557407447424 T^{6} + T^{8} \)
$61$ \( ( \)\(10\!\cdots\!96\)\( - \)\(19\!\cdots\!92\)\( T + 317957233175192748 T^{2} - 592019372 T^{3} + T^{4} )^{2} \)
$67$ \( ( \)\(12\!\cdots\!16\)\( + \)\(52\!\cdots\!04\)\( T + 372314373353235372 T^{2} - 148682924 T^{3} + T^{4} )^{2} \)
$71$ \( ( \)\(49\!\cdots\!00\)\( + 1847488216292328960 T^{2} + T^{4} )^{2} \)
$73$ \( ( 2363496913262627140 - 3267134500 T + T^{2} )^{4} \)
$79$ \( ( \)\(13\!\cdots\!76\)\( - \)\(36\!\cdots\!84\)\( T + 3678893373957736332 T^{2} + 99641284 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(32\!\cdots\!36\)\( - \)\(34\!\cdots\!76\)\( T^{2} + \)\(29\!\cdots\!72\)\( T^{4} - 5977139602070968704 T^{6} + T^{8} \)
$89$ \( ( \)\(15\!\cdots\!84\)\( + 27084125311735371264 T^{2} + T^{4} )^{2} \)
$97$ \( ( \)\(90\!\cdots\!16\)\( - \)\(18\!\cdots\!72\)\( T + \)\(28\!\cdots\!28\)\( T^{2} - 19588177532 T^{3} + T^{4} )^{2} \)
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