Properties

Label 1296.5.e.g
Level 1296
Weight 5
Character orbit 1296.e
Analytic conductor 133.967
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(133.967472157\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{20} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( 3 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 3 - \beta_{3} ) q^{7} + \beta_{2} q^{11} + ( 2 + \beta_{3} + \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{17} + ( -70 - \beta_{6} ) q^{19} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{23} + ( -88 + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{2} + \beta_{4} + \beta_{7} ) q^{29} + ( 46 + 4 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{31} + ( 15 \beta_{1} - 6 \beta_{2} + \beta_{4} - 2 \beta_{7} ) q^{35} + ( -5 - 19 \beta_{3} - 5 \beta_{5} - \beta_{6} ) q^{37} + ( 26 \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{41} + ( -15 + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{43} + ( -11 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} - 4 \beta_{7} ) q^{47} + ( 71 - 19 \beta_{3} - 11 \beta_{5} + 2 \beta_{6} ) q^{49} + ( 61 \beta_{1} - 25 \beta_{2} + 5 \beta_{4} + 5 \beta_{7} ) q^{53} + ( 212 - 34 \beta_{3} - 13 \beta_{5} + \beta_{6} ) q^{55} + ( 63 \beta_{1} + 20 \beta_{2} + \beta_{4} - 7 \beta_{7} ) q^{59} + ( 480 - 53 \beta_{3} + 19 \beta_{5} + 2 \beta_{6} ) q^{61} + ( 106 \beta_{1} + 41 \beta_{2} + 3 \beta_{4} + 9 \beta_{7} ) q^{65} + ( 16 - 96 \beta_{3} + 3 \beta_{5} ) q^{67} + ( -35 \beta_{1} - 17 \beta_{2} - 9 \beta_{4} - 13 \beta_{7} ) q^{71} + ( -953 + 83 \beta_{3} - \beta_{5} - 8 \beta_{6} ) q^{73} + ( -105 \beta_{1} + 18 \beta_{2} - 14 \beta_{4} + 6 \beta_{7} ) q^{77} + ( -568 - 90 \beta_{3} + 3 \beta_{5} - 7 \beta_{6} ) q^{79} + ( 229 \beta_{1} + 12 \beta_{2} + 7 \beta_{4} - 12 \beta_{7} ) q^{83} + ( 731 - 97 \beta_{3} - 7 \beta_{5} + 7 \beta_{6} ) q^{85} + ( -241 \beta_{1} - 19 \beta_{2} + 3 \beta_{4} + 7 \beta_{7} ) q^{89} + ( -1940 + 162 \beta_{3} - 7 \beta_{5} - 7 \beta_{6} ) q^{91} + ( -384 \beta_{1} + 14 \beta_{2} - 20 \beta_{4} - 16 \beta_{7} ) q^{95} + ( -1822 - 17 \beta_{3} + \beta_{5} - 9 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 26q^{7} + O(q^{10}) \) \( 8q + 26q^{7} + 10q^{13} - 562q^{19} - 706q^{25} + 374q^{31} + 16q^{37} - 136q^{43} + 654q^{49} + 1818q^{55} + 3874q^{61} + 308q^{67} - 7802q^{73} - 4390q^{79} + 6084q^{85} - 15830q^{91} - 14564q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 6 x^{6} + 121 x^{5} + 1104 x^{4} - 1647 x^{3} + 6529 x^{2} + 85254 x + 440076\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 17 \nu^{7} - 142 \nu^{6} + 635 \nu^{5} - 1115 \nu^{4} + 17923 \nu^{3} - 159143 \nu^{2} + 464580 \nu - 386178 \)\()/104247\)
\(\beta_{2}\)\(=\)\((\)\( 53 \nu^{7} - 367 \nu^{6} + 5765 \nu^{5} - 70703 \nu^{4} + 113944 \nu^{3} + 106240 \nu^{2} - 1050579 \nu - 42441945 \)\()/104247\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} + 34 \nu^{5} + 116 \nu^{4} + 197 \nu^{3} - 223 \nu^{2} + 25974 \nu + 60657 \)\()/729\)
\(\beta_{4}\)\(=\)\((\)\( 118 \nu^{7} + 49 \nu^{6} - 1775 \nu^{5} + 11717 \nu^{4} + 118754 \nu^{3} + 166664 \nu^{2} + 172422 \nu + 2470650 \)\()/34749\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{7} - 38 \nu^{6} + 73 \nu^{5} + 410 \nu^{4} + 428 \nu^{3} - 18433 \nu^{2} + 18954 \nu + 178665 \)\()/729\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 2 \nu^{6} + 143 \nu^{5} + 661 \nu^{4} + 4441 \nu^{3} + 15841 \nu^{2} + 96174 \nu + 445521 \)\()/729\)
\(\beta_{7}\)\(=\)\((\)\( -1052 \nu^{7} - 1433 \nu^{6} + 40423 \nu^{5} - 237610 \nu^{4} - 608323 \nu^{3} + 281948 \nu^{2} + 16720785 \nu - 71046885 \)\()/104247\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 18 \beta_{3} - 3 \beta_{2} + 22 \beta_{1} + 186\)\()/486\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{7} + 3 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - \beta_{2} - 116 \beta_{1} - 60\)\()/162\)
\(\nu^{3}\)\(=\)\((\)\(19 \beta_{7} + 24 \beta_{6} + 27 \beta_{5} + 56 \beta_{4} - 228 \beta_{3} - 37 \beta_{2} + 125 \beta_{1} - 7947\)\()/162\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{7} + 111 \beta_{6} + 96 \beta_{5} - 64 \beta_{4} - 570 \beta_{3} - 220 \beta_{2} - 219 \beta_{1} - 120225\)\()/162\)
\(\nu^{5}\)\(=\)\((\)\(-217 \beta_{7} + 930 \beta_{6} + 285 \beta_{5} - 2155 \beta_{4} - 2505 \beta_{3} + 515 \beta_{2} + 2208 \beta_{1} - 206598\)\()/162\)
\(\nu^{6}\)\(=\)\((\)\(-4159 \beta_{7} + 651 \beta_{6} - 6591 \beta_{5} - 13543 \beta_{4} + 15891 \beta_{3} + 5291 \beta_{2} + 68865 \beta_{1} + 432933\)\()/162\)
\(\nu^{7}\)\(=\)\((\)\(-26773 \beta_{7} - 19995 \beta_{6} - 32457 \beta_{5} - 47548 \beta_{4} + 237246 \beta_{3} + 67505 \beta_{2} + 53688 \beta_{1} + 13250454\)\()/162\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−3.41053 2.74723i
−3.05006 3.25531i
4.23522 + 4.06612i
3.72537 + 4.42407i
3.72537 4.42407i
4.23522 4.06612i
−3.05006 + 3.25531i
−3.41053 + 2.74723i
0 0 0 40.2664i 0 −14.7738 0 0 0
161.2 0 0 0 31.6564i 0 75.3660 0 0 0
161.3 0 0 0 12.2819i 0 14.2840 0 0 0
161.4 0 0 0 8.86801i 0 −61.8763 0 0 0
161.5 0 0 0 8.86801i 0 −61.8763 0 0 0
161.6 0 0 0 12.2819i 0 14.2840 0 0 0
161.7 0 0 0 31.6564i 0 75.3660 0 0 0
161.8 0 0 0 40.2664i 0 −14.7738 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.5.e.g 8
3.b odd 2 1 inner 1296.5.e.g 8
4.b odd 2 1 324.5.c.a 8
9.c even 3 1 144.5.q.c 8
9.c even 3 1 432.5.q.c 8
9.d odd 6 1 144.5.q.c 8
9.d odd 6 1 432.5.q.c 8
12.b even 2 1 324.5.c.a 8
36.f odd 6 1 36.5.g.a 8
36.f odd 6 1 108.5.g.a 8
36.h even 6 1 36.5.g.a 8
36.h even 6 1 108.5.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 36.f odd 6 1
36.5.g.a 8 36.h even 6 1
108.5.g.a 8 36.f odd 6 1
108.5.g.a 8 36.h even 6 1
144.5.q.c 8 9.c even 3 1
144.5.q.c 8 9.d odd 6 1
324.5.c.a 8 4.b odd 2 1
324.5.c.a 8 12.b even 2 1
432.5.q.c 8 9.c even 3 1
432.5.q.c 8 9.d odd 6 1
1296.5.e.g 8 1.a even 1 1 trivial
1296.5.e.g 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2853 T_{5}^{6} + 2238759 T_{5}^{4} + 403999407 T_{5}^{2} + 19274879556 \) acting on \(S_{5}^{\mathrm{new}}(1296, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 2147 T^{2} + 2477509 T^{4} - 2147976218 T^{6} + 1511855308306 T^{8} - 839053210156250 T^{10} + 378037872314453125 T^{12} - \)\(12\!\cdots\!75\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} \)
$7$ \( ( 1 - 13 T + 4723 T^{2} - 93076 T^{3} + 12134350 T^{4} - 223475476 T^{5} + 27227155123 T^{6} - 179936733613 T^{7} + 33232930569601 T^{8} )^{2} \)
$11$ \( 1 - 67376 T^{2} + 2511150250 T^{4} - 60712318203824 T^{6} + 1049763807787064539 T^{8} - \)\(13\!\cdots\!44\)\( T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(66\!\cdots\!16\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} \)
$13$ \( ( 1 - 5 T + 42079 T^{2} - 3738530 T^{3} + 1399773016 T^{4} - 106776155330 T^{5} + 34325133008959 T^{6} - 116490425612405 T^{7} + 665416609183179841 T^{8} )^{2} \)
$17$ \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(25\!\cdots\!74\)\( T^{12} - \)\(97\!\cdots\!25\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 + 281 T + 110170 T^{2} - 68843041 T^{3} - 21846246566 T^{4} - 8971693946161 T^{5} + 1871079140226970 T^{6} + 621941492257591241 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 - 103955 T^{2} + 204060092029 T^{4} - 14985633296690750 T^{6} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{10} + \)\(12\!\cdots\!69\)\( T^{12} - \)\(49\!\cdots\!55\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - 3708803 T^{2} + 6181658247973 T^{4} - 6420936064399980986 T^{6} + \)\(50\!\cdots\!30\)\( T^{8} - \)\(32\!\cdots\!46\)\( T^{10} + \)\(15\!\cdots\!33\)\( T^{12} - \)\(46\!\cdots\!43\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( ( 1 - 187 T + 2550973 T^{2} - 331939744 T^{3} + 3082958795560 T^{4} - 306553324318624 T^{5} + 2175702008453980093 T^{6} - \)\(14\!\cdots\!07\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} )^{2} \)
$37$ \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \)
$41$ \( 1 - 12649388 T^{2} + 74446575673498 T^{4} - \)\(29\!\cdots\!76\)\( T^{6} + \)\(89\!\cdots\!55\)\( T^{8} - \)\(23\!\cdots\!96\)\( T^{10} + \)\(47\!\cdots\!18\)\( T^{12} - \)\(64\!\cdots\!68\)\( T^{14} + \)\(40\!\cdots\!81\)\( T^{16} \)
$43$ \( ( 1 + 68 T + 12955228 T^{2} + 545371076 T^{3} + 65226299853025 T^{4} + 1864515179999876 T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \)
$47$ \( 1 - 19981955 T^{2} + 227810016714829 T^{4} - \)\(17\!\cdots\!90\)\( T^{6} + \)\(10\!\cdots\!46\)\( T^{8} - \)\(42\!\cdots\!90\)\( T^{10} + \)\(12\!\cdots\!09\)\( T^{12} - \)\(26\!\cdots\!55\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 31340696 T^{2} + 646808461063090 T^{4} - \)\(10\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!19\)\( T^{8} - \)\(15\!\cdots\!24\)\( T^{10} + \)\(13\!\cdots\!90\)\( T^{12} - \)\(99\!\cdots\!56\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( ( 1 - 1937 T + 14281603 T^{2} + 60991858006 T^{3} - 94182960283700 T^{4} + 844483568245653046 T^{5} + \)\(27\!\cdots\!43\)\( T^{6} - \)\(51\!\cdots\!77\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \)
$67$ \( ( 1 - 154 T + 33859570 T^{2} + 10195927724 T^{3} + 608674748042419 T^{4} + 205459373273578604 T^{5} + \)\(13\!\cdots\!70\)\( T^{6} - \)\(12\!\cdots\!94\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} )^{2} \)
$71$ \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(73\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \)
$73$ \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(89\!\cdots\!21\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \)
$79$ \( ( 1 + 2195 T + 92542939 T^{2} + 195388180760 T^{3} + 5006999286743146 T^{4} + 7610385467044641560 T^{5} + \)\(14\!\cdots\!79\)\( T^{6} + \)\(12\!\cdots\!95\)\( T^{7} + \)\(23\!\cdots\!21\)\( T^{8} )^{2} \)
$83$ \( 1 - 81188291 T^{2} + 8014262915096365 T^{4} - \)\(34\!\cdots\!54\)\( T^{6} + \)\(22\!\cdots\!34\)\( T^{8} - \)\(78\!\cdots\!14\)\( T^{10} + \)\(40\!\cdots\!65\)\( T^{12} - \)\(92\!\cdots\!11\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!36\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( ( 1 + 7282 T + 336254488 T^{2} + 1720884578884 T^{3} + 43500636050571385 T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(26\!\cdots\!68\)\( T^{6} + \)\(50\!\cdots\!62\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \)
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