Properties

Label 1764.3.z.n
Level $1764$
Weight $3$
Character orbit 1764.z
Analytic conductor $48.066$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 12 x^{14} + 114 x^{12} - 336 x^{10} + 755 x^{8} - 336 x^{6} + 114 x^{4} - 12 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{12} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{4} - \beta_{12} ) q^{5} +O(q^{10})\) \( q + ( -\beta_{4} - \beta_{12} ) q^{5} + \beta_{11} q^{11} + \beta_{13} q^{13} -3 \beta_{4} q^{17} + ( \beta_{3} - 2 \beta_{9} + 2 \beta_{13} ) q^{19} + ( \beta_{1} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{23} + ( 3 + 3 \beta_{2} - \beta_{8} ) q^{25} + ( \beta_{5} - \beta_{10} ) q^{29} + ( -5 \beta_{7} - 4 \beta_{9} ) q^{31} + ( 16 \beta_{2} - 3 \beta_{6} + 3 \beta_{8} ) q^{37} + ( \beta_{4} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{41} + ( 22 - 2 \beta_{6} ) q^{43} + ( -2 \beta_{4} - 2 \beta_{12} - \beta_{15} ) q^{47} -2 \beta_{11} q^{53} + ( \beta_{3} - \beta_{7} + 18 \beta_{13} ) q^{55} + ( 3 \beta_{4} + \beta_{14} ) q^{59} + ( 12 \beta_{3} - 13 \beta_{9} + 13 \beta_{13} ) q^{61} + ( -\beta_{5} + \beta_{11} ) q^{65} + ( 32 + 32 \beta_{2} - 6 \beta_{8} ) q^{67} + ( \beta_{5} + 2 \beta_{10} ) q^{71} + ( -22 \beta_{7} - 21 \beta_{9} ) q^{73} + ( 54 \beta_{2} - 8 \beta_{6} + 8 \beta_{8} ) q^{79} + ( -2 \beta_{4} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 84 - 3 \beta_{6} ) q^{85} + ( 9 \beta_{4} + 9 \beta_{12} + 2 \beta_{15} ) q^{89} + ( \beta_{1} + 2 \beta_{11} ) q^{95} + ( -2 \beta_{3} + 2 \beta_{7} + 27 \beta_{13} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 24q^{25} - 128q^{37} + 352q^{43} + 256q^{67} - 432q^{79} + 1344q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 12 x^{14} + 114 x^{12} - 336 x^{10} + 755 x^{8} - 336 x^{6} + 114 x^{4} - 12 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -473 \nu^{14} - 96474 \nu^{12} + 1132373 \nu^{10} - 10999257 \nu^{8} + 29348105 \nu^{6} - 62719633 \nu^{4} + 4563258 \nu^{2} - 481239 \)\()/157522\)
\(\beta_{2}\)\(=\)\((\)\( -8724 \nu^{14} + 103818 \nu^{12} - 984456 \nu^{10} + 2836231 \nu^{8} - 6333624 \nu^{6} + 2379102 \nu^{4} - 955212 \nu^{2} + 21780 \)\()/78761\)
\(\beta_{3}\)\(=\)\((\)\( -7388 \nu^{15} + 75216 \nu^{13} - 686249 \nu^{11} + 1014272 \nu^{9} - 1669588 \nu^{7} - 5856677 \nu^{5} - 234744 \nu^{3} + 24592 \nu \)\()/78761\)
\(\beta_{4}\)\(=\)\((\)\( -32730 \nu^{15} + 379529 \nu^{13} - 3575207 \nu^{11} + 9517884 \nu^{9} - 20532779 \nu^{7} + 1479396 \nu^{5} - 156013 \nu^{3} - 3968033 \nu \)\()/315044\)
\(\beta_{5}\)\(=\)\((\)\( 125820 \nu^{14} - 1459483 \nu^{12} + 13743738 \nu^{10} - 36588456 \nu^{8} + 78586038 \nu^{6} - 5687064 \nu^{4} + 599742 \nu^{2} + 3033171 \)\()/157522\)
\(\beta_{6}\)\(=\)\((\)\( 210 \nu^{14} - 2436 \nu^{12} + 22939 \nu^{10} - 61068 \nu^{8} + 131117 \nu^{6} - 9492 \nu^{4} + 1001 \nu^{2} + 2310 \)\()/226\)
\(\beta_{7}\)\(=\)\((\)\( -34350 \nu^{15} + 398539 \nu^{13} - 3752165 \nu^{11} + 9988980 \nu^{9} - 21394552 \nu^{7} + 1552620 \nu^{5} - 163735 \nu^{3} + 92422 \nu \)\()/78761\)
\(\beta_{8}\)\(=\)\((\)\( -191009 \nu^{14} + 2326128 \nu^{12} - 22169357 \nu^{10} + 67895142 \nu^{8} - 154104811 \nu^{6} + 85598856 \nu^{4} - 23312730 \nu^{2} + 2454270 \)\()/157522\)
\(\beta_{9}\)\(=\)\((\)\( -186810 \nu^{15} + 2166991 \nu^{13} - 20405879 \nu^{11} + 54324348 \nu^{9} - 116648235 \nu^{7} + 8443812 \nu^{5} - 890461 \nu^{3} - 4213215 \nu \)\()/315044\)
\(\beta_{10}\)\(=\)\((\)\( 274170 \nu^{14} - 3181229 \nu^{12} + 29948503 \nu^{10} - 79728636 \nu^{8} + 170614699 \nu^{6} - 12392484 \nu^{4} + 1306877 \nu^{2} - 206391 \)\()/157522\)
\(\beta_{11}\)\(=\)\((\)\( -346926 \nu^{14} + 4174002 \nu^{12} - 39675882 \nu^{10} + 117756687 \nu^{8} - 265095942 \nu^{6} + 123410351 \nu^{4} - 40041792 \nu^{2} + 4215021 \)\()/157522\)
\(\beta_{12}\)\(=\)\((\)\( 11593 \nu^{15} - 137995 \nu^{13} + 1308540 \nu^{11} - 3771965 \nu^{9} + 8418660 \nu^{7} - 3162305 \nu^{5} + 1198319 \nu^{3} - 28950 \nu \)\()/7684\)
\(\beta_{13}\)\(=\)\((\)\( -39393 \nu^{15} + 469469 \nu^{13} - 4451748 \nu^{11} + 12864843 \nu^{9} - 28640892 \nu^{7} + 10758391 \nu^{5} - 3342383 \nu^{3} + 98490 \nu \)\()/18532\)
\(\beta_{14}\)\(=\)\((\)\( -3450630 \nu^{15} + 40028431 \nu^{13} - 376923817 \nu^{11} + 1003443204 \nu^{9} - 2153783649 \nu^{7} + 155968476 \nu^{5} - 16448003 \nu^{3} - 52231323 \nu \)\()/315044\)
\(\beta_{15}\)\(=\)\((\)\( -39787 \nu^{15} + 481194 \nu^{13} - 4579212 \nu^{11} + 13778057 \nu^{9} - 31129685 \nu^{7} + 15713328 \nu^{5} - 4705218 \nu^{3} + 495319 \nu \)\()/1921\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} + 28 \beta_{9} - 7 \beta_{7} - 25 \beta_{4}\)\()/98\)
\(\nu^{2}\)\(=\)\((\)\(11 \beta_{11} + \beta_{10} - 7 \beta_{8} + 7 \beta_{6} - 11 \beta_{5} - 147 \beta_{2} + \beta_{1}\)\()/49\)
\(\nu^{3}\)\(=\)\((\)\(-13 \beta_{15} - 13 \beta_{14} + 280 \beta_{13} + 116 \beta_{12} - 21 \beta_{7} - 13 \beta_{4} + 21 \beta_{3}\)\()/98\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{11} - 12 \beta_{8} - 147 \beta_{2} - 147\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(-125 \beta_{15} + 2464 \beta_{13} + 844 \beta_{12} - 2464 \beta_{9} + 844 \beta_{4} - 21 \beta_{3}\)\()/98\)
\(\nu^{6}\)\(=\)\((\)\(29 \beta_{10} - 791 \beta_{6} + 857 \beta_{5} - 8379\)\()/49\)
\(\nu^{7}\)\(=\)\((\)\(1121 \beta_{14} - 21532 \beta_{9} - 791 \beta_{7} + 8033 \beta_{4}\)\()/98\)
\(\nu^{8}\)\(=\)\((\)\(-1068 \beta_{11} + 48 \beta_{10} + 1008 \beta_{8} - 1008 \beta_{6} + 1068 \beta_{5} + 10199 \beta_{2} + 48 \beta_{1}\)\()/7\)
\(\nu^{9}\)\(=\)\((\)\(9857 \beta_{15} + 9857 \beta_{14} - 187796 \beta_{13} - 58992 \beta_{12} - 8617 \beta_{7} + 9857 \beta_{4} + 8617 \beta_{3}\)\()/98\)
\(\nu^{10}\)\(=\)\((\)\(-65167 \beta_{11} + 61943 \beta_{8} + 617547 \beta_{2} + 3163 \beta_{1} + 617547\)\()/49\)
\(\nu^{11}\)\(=\)\((\)\(86141 \beta_{15} - 1636880 \beta_{13} - 510556 \beta_{12} + 1636880 \beta_{9} - 510556 \beta_{4} + 79947 \beta_{3}\)\()/98\)
\(\nu^{12}\)\(=\)\((\)\(-4032 \beta_{10} + 77292 \beta_{6} - 81130 \beta_{5} + 766899\)\()/7\)
\(\nu^{13}\)\(=\)\((\)\(-751309 \beta_{14} + 14264600 \beta_{9} + 710325 \beta_{7} - 5190321 \beta_{4}\)\()/98\)
\(\nu^{14}\)\(=\)\((\)\(4948765 \beta_{11} - 247801 \beta_{10} - 4718119 \beta_{8} + 4718119 \beta_{6} - 4948765 \beta_{5} - 46741443 \beta_{2} - 247801 \beta_{1}\)\()/49\)
\(\nu^{15}\)\(=\)\((\)\(-6548641 \beta_{15} - 6548641 \beta_{14} + 124300820 \beta_{13} + 38652456 \beta_{12} + 6228103 \beta_{7} - 6548641 \beta_{4} - 6228103 \beta_{3}\)\()/98\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
−1.45341 0.839125i
−0.516029 0.297929i
−2.55641 1.47595i
−0.293380 0.169383i
2.55641 + 1.47595i
0.293380 + 0.169383i
1.45341 + 0.839125i
0.516029 + 0.297929i
−1.45341 + 0.839125i
−0.516029 + 0.297929i
−2.55641 + 1.47595i
−0.293380 + 0.169383i
2.55641 1.47595i
0.293380 0.169383i
1.45341 0.839125i
0.516029 0.297929i
0 0 0 −5.33147 + 3.07813i 0 0 0 0 0
325.2 0 0 0 −5.33147 + 3.07813i 0 0 0 0 0
325.3 0 0 0 −3.68448 + 2.12723i 0 0 0 0 0
325.4 0 0 0 −3.68448 + 2.12723i 0 0 0 0 0
325.5 0 0 0 3.68448 2.12723i 0 0 0 0 0
325.6 0 0 0 3.68448 2.12723i 0 0 0 0 0
325.7 0 0 0 5.33147 3.07813i 0 0 0 0 0
325.8 0 0 0 5.33147 3.07813i 0 0 0 0 0
901.1 0 0 0 −5.33147 3.07813i 0 0 0 0 0
901.2 0 0 0 −5.33147 3.07813i 0 0 0 0 0
901.3 0 0 0 −3.68448 2.12723i 0 0 0 0 0
901.4 0 0 0 −3.68448 2.12723i 0 0 0 0 0
901.5 0 0 0 3.68448 + 2.12723i 0 0 0 0 0
901.6 0 0 0 3.68448 + 2.12723i 0 0 0 0 0
901.7 0 0 0 5.33147 + 3.07813i 0 0 0 0 0
901.8 0 0 0 5.33147 + 3.07813i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.z.n 16
3.b odd 2 1 inner 1764.3.z.n 16
7.b odd 2 1 inner 1764.3.z.n 16
7.c even 3 1 1764.3.d.g 8
7.c even 3 1 inner 1764.3.z.n 16
7.d odd 6 1 1764.3.d.g 8
7.d odd 6 1 inner 1764.3.z.n 16
21.c even 2 1 inner 1764.3.z.n 16
21.g even 6 1 1764.3.d.g 8
21.g even 6 1 inner 1764.3.z.n 16
21.h odd 6 1 1764.3.d.g 8
21.h odd 6 1 inner 1764.3.z.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.d.g 8 7.c even 3 1
1764.3.d.g 8 7.d odd 6 1
1764.3.d.g 8 21.g even 6 1
1764.3.d.g 8 21.h odd 6 1
1764.3.z.n 16 1.a even 1 1 trivial
1764.3.z.n 16 3.b odd 2 1 inner
1764.3.z.n 16 7.b odd 2 1 inner
1764.3.z.n 16 7.c even 3 1 inner
1764.3.z.n 16 7.d odd 6 1 inner
1764.3.z.n 16 21.c even 2 1 inner
1764.3.z.n 16 21.g even 6 1 inner
1764.3.z.n 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} - 56 T_{5}^{6} + 2450 T_{5}^{4} - 38416 T_{5}^{2} + 470596 \)
\( T_{11}^{8} + 364 T_{11}^{6} + 131124 T_{11}^{4} + 499408 T_{11}^{2} + 1882384 \)
\( T_{13}^{4} + 20 T_{13}^{2} + 2 \)
\( T_{19}^{8} - 136 T_{19}^{6} + 14264 T_{19}^{4} - 575552 T_{19}^{2} + 17909824 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 470596 - 38416 T^{2} + 2450 T^{4} - 56 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 1882384 + 499408 T^{2} + 131124 T^{4} + 364 T^{6} + T^{8} )^{2} \)
$13$ \( ( 2 + 20 T^{2} + T^{4} )^{4} \)
$17$ \( ( 3087580356 - 28005264 T^{2} + 198450 T^{4} - 504 T^{6} + T^{8} )^{2} \)
$19$ \( ( 17909824 - 575552 T^{2} + 14264 T^{4} - 136 T^{6} + T^{8} )^{2} \)
$23$ \( ( 157218594064 + 655031216 T^{2} + 2332596 T^{4} + 1652 T^{6} + T^{8} )^{2} \)
$29$ \( ( 725788 - 2100 T^{2} + T^{4} )^{4} \)
$31$ \( ( 49778964544 - 258809920 T^{2} + 1122488 T^{4} - 1160 T^{6} + T^{8} )^{2} \)
$37$ \( ( 391876 - 20032 T + 1650 T^{2} + 32 T^{3} + T^{4} )^{4} \)
$41$ \( ( 3655694 + 4928 T^{2} + T^{4} )^{4} \)
$43$ \( ( 92 - 44 T + T^{2} )^{8} \)
$47$ \( ( 36741733758016 - 29871052288 T^{2} + 18223688 T^{4} - 4928 T^{6} + T^{8} )^{2} \)
$53$ \( ( 481890304 + 31962112 T^{2} + 2097984 T^{4} + 1456 T^{6} + T^{8} )^{2} \)
$59$ \( ( 6953684616256 - 13585741568 T^{2} + 23906120 T^{4} - 5152 T^{6} + T^{8} )^{2} \)
$61$ \( ( 654840832460164 - 265827444904 T^{2} + 82320686 T^{4} - 10388 T^{6} + T^{8} )^{2} \)
$67$ \( ( 6270016 + 160256 T + 6600 T^{2} - 64 T^{3} + T^{4} )^{4} \)
$71$ \( ( 1318492 - 5964 T^{2} + T^{4} )^{4} \)
$73$ \( ( 15042807127370884 - 3002941103048 T^{2} + 476817134 T^{4} - 24484 T^{6} + T^{8} )^{2} \)
$79$ \( ( 11262736 - 362448 T + 15020 T^{2} + 108 T^{3} + T^{4} )^{4} \)
$83$ \( ( 86940896 + 19264 T^{2} + T^{4} )^{4} \)
$89$ \( ( 10999052145365956 - 2390342133872 T^{2} + 414598898 T^{4} - 22792 T^{6} + T^{8} )^{2} \)
$97$ \( ( 235298 + 14308 T^{2} + T^{4} )^{4} \)
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