Properties

Label 1764.4.t.a
Level $1764$
Weight $4$
Character orbit 1764.t
Analytic conductor $104.079$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 42 x^{12} + 1139 x^{8} + 26250 x^{4} + 390625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{5} +O(q^{10})\) \( q + \beta_{11} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{11} + ( \beta_{12} - \beta_{14} ) q^{13} + ( \beta_{7} + \beta_{13} - \beta_{15} ) q^{17} + ( -3 \beta_{9} + 2 \beta_{10} - 2 \beta_{14} ) q^{19} + ( 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{23} + ( -5 \beta_{1} - 4 \beta_{8} ) q^{25} + ( 10 \beta_{4} + \beta_{6} ) q^{29} + ( 6 \beta_{9} - 6 \beta_{12} ) q^{31} + ( -64 - 64 \beta_{1} + 6 \beta_{5} ) q^{37} + ( 29 \beta_{7} - 29 \beta_{11} - 3 \beta_{13} ) q^{41} + ( 4 - 5 \beta_{5} + 5 \beta_{8} ) q^{43} + ( -32 \beta_{11} + 3 \beta_{15} ) q^{47} + ( -18 \beta_{2} - 13 \beta_{3} - 18 \beta_{4} ) q^{53} + ( -7 \beta_{12} - 4 \beta_{14} ) q^{55} + ( -12 \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{59} + ( -20 \beta_{9} + 10 \beta_{10} - 10 \beta_{14} ) q^{61} + ( 8 \beta_{2} - 24 \beta_{3} - 24 \beta_{6} ) q^{65} + ( -292 \beta_{1} + 4 \beta_{8} ) q^{67} + ( -19 \beta_{4} - 10 \beta_{6} ) q^{71} + ( -3 \beta_{9} - 11 \beta_{10} + 3 \beta_{12} ) q^{73} + ( -224 - 224 \beta_{1} - 12 \beta_{5} ) q^{79} + ( -28 \beta_{7} + 28 \beta_{11} + 9 \beta_{13} ) q^{83} + ( -168 - 4 \beta_{5} + 4 \beta_{8} ) q^{85} + ( 19 \beta_{11} - 5 \beta_{15} ) q^{89} + ( 56 \beta_{2} - 88 \beta_{3} + 56 \beta_{4} ) q^{95} + ( -\beta_{12} - 15 \beta_{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 40q^{25} - 512q^{37} + 64q^{43} + 2336q^{67} - 1792q^{79} - 2688q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 42 x^{12} + 1139 x^{8} + 26250 x^{4} + 390625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 42 \nu^{12} + 1139 \nu^{8} + 47838 \nu^{4} + 390625 \)\()/711875\)
\(\beta_{2}\)\(=\)\((\)\( -771 \nu^{12} - 71757 \nu^{8} - 878169 \nu^{4} - 20238750 \)\()/1423750\)
\(\beta_{3}\)\(=\)\((\)\( 801 \nu^{14} + 174267 \nu^{10} + 912339 \nu^{6} + 21026250 \nu^{2} \)\()/35593750\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{12} - 6993 \)\()/2278\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{14} - 150189 \nu^{2} \)\()/28475\)
\(\beta_{6}\)\(=\)\((\)\( -378 \nu^{14} - 10251 \nu^{10} - 53667 \nu^{6} - 3515625 \nu^{2} \)\()/2093750\)
\(\beta_{7}\)\(=\)\((\)\( -1028 \nu^{15} - 21000 \nu^{13} - 95676 \nu^{11} - 569500 \nu^{9} - 2594642 \nu^{7} - 23919000 \nu^{5} - 86782500 \nu^{3} - 373281250 \nu \)\()/88984375\)
\(\beta_{8}\)\(=\)\((\)\( -6567 \nu^{14} - 228939 \nu^{10} - 7479813 \nu^{6} - 172383750 \nu^{2} \)\()/17796875\)
\(\beta_{9}\)\(=\)\((\)\( -2861 \nu^{15} - 7600 \nu^{13} - 94537 \nu^{11} - 1053575 \nu^{9} - 4682429 \nu^{7} - 26453275 \nu^{5} + 43070000 \nu^{3} - 857984375 \nu \)\()/88984375\)
\(\beta_{10}\)\(=\)\((\)\( -3304 \nu^{15} - 1850 \nu^{13} + 384982 \nu^{11} + 797300 \nu^{9} + 6202994 \nu^{7} - 37700900 \nu^{5} + 153883750 \nu^{3} - 297718750 \nu \)\()/88984375\)
\(\beta_{11}\)\(=\)\((\)\( 4778 \nu^{15} - 10500 \nu^{13} + 95676 \nu^{11} - 284750 \nu^{9} + 2594642 \nu^{7} - 11959500 \nu^{5} + 5827500 \nu^{3} - 453593750 \nu \)\()/88984375\)
\(\beta_{12}\)\(=\)\((\)\( -7597 \nu^{15} + 54475 \nu^{13} - 189074 \nu^{11} + 1053575 \nu^{9} - 9364858 \nu^{7} + 26453275 \nu^{5} - 140335625 \nu^{3} + 113000000 \nu \)\()/88984375\)
\(\beta_{13}\)\(=\)\((\)\( -734 \nu^{15} + 13050 \nu^{13} - 45828 \nu^{11} + 204350 \nu^{9} + 252724 \nu^{7} + 2301450 \nu^{5} - 4946250 \nu^{3} + 87125000 \nu \)\()/5234375\)
\(\beta_{14}\)\(=\)\((\)\( 19642 \nu^{15} + 45950 \nu^{13} + 769964 \nu^{11} + 398650 \nu^{9} + 12405988 \nu^{7} - 18850450 \nu^{5} + 274988750 \nu^{3} + 707875000 \nu \)\()/88984375\)
\(\beta_{15}\)\(=\)\((\)\( -30614 \nu^{15} + 59400 \nu^{13} - 389538 \nu^{11} - 3473950 \nu^{9} + 2148154 \nu^{7} - 39124650 \nu^{5} - 316695000 \nu^{3} - 611968750 \nu \)\()/88984375\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{14} - 2 \beta_{12} - 12 \beta_{11} - \beta_{10} - 2 \beta_{9} + 6 \beta_{7}\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{15} - \beta_{14} + 2 \beta_{13} - 14 \beta_{12} - 26 \beta_{11} + 2 \beta_{10} + 28 \beta_{9} - 26 \beta_{7}\)\()/72\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{4} + 2 \beta_{2} + 63 \beta_{1}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(10 \beta_{15} - 37 \beta_{14} - 20 \beta_{13} + 28 \beta_{12} + 122 \beta_{11} - 37 \beta_{10} - 14 \beta_{9} - 244 \beta_{7}\)\()/72\)
\(\nu^{6}\)\(=\)\((\)\(-51 \beta_{8} + 134 \beta_{6} + 51 \beta_{5}\)\()/18\)
\(\nu^{7}\)\(=\)\((\)\(84 \beta_{15} - 166 \beta_{14} + 84 \beta_{13} - 338 \beta_{12} + 684 \beta_{11} + 83 \beta_{10} - 338 \beta_{9} - 342 \beta_{7}\)\()/72\)
\(\nu^{8}\)\(=\)\(-28 \beta_{2} - 257 \beta_{1} - 257\)
\(\nu^{9}\)\(=\)\((\)\(-840 \beta_{15} - 929 \beta_{14} + 420 \beta_{13} + 662 \beta_{12} + 1374 \beta_{11} + 1858 \beta_{10} - 1324 \beta_{9} + 1374 \beta_{7}\)\()/72\)
\(\nu^{10}\)\(=\)\((\)\(267 \beta_{8} + 4378 \beta_{3}\)\()/18\)
\(\nu^{11}\)\(=\)\((\)\(2278 \beta_{15} + 4111 \beta_{14} - 4556 \beta_{13} + 10892 \beta_{12} + 1886 \beta_{11} + 4111 \beta_{10} - 5446 \beta_{9} - 3772 \beta_{7}\)\()/72\)
\(\nu^{12}\)\(=\)\((\)\(-2278 \beta_{4} - 6993\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(11390 \beta_{15} + 31786 \beta_{14} + 11390 \beta_{13} + 36554 \beta_{12} + 37084 \beta_{11} - 15893 \beta_{10} + 36554 \beta_{9} - 18542 \beta_{7}\)\()/72\)
\(\nu^{14}\)\(=\)\((\)\(-100126 \beta_{6} + 20661 \beta_{5} - 100126 \beta_{3}\)\()/18\)
\(\nu^{15}\)\(=\)\((\)\(-86352 \beta_{15} + 120787 \beta_{14} + 43176 \beta_{13} - 17482 \beta_{12} + 292962 \beta_{11} - 241574 \beta_{10} + 34964 \beta_{9} + 292962 \beta_{7}\)\()/72\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
2.07637 + 0.829861i
1.75687 + 1.38326i
−0.829861 + 2.07637i
1.38326 1.75687i
0.829861 2.07637i
−1.38326 + 1.75687i
−2.07637 0.829861i
−1.75687 1.38326i
2.07637 0.829861i
1.75687 1.38326i
−0.829861 2.07637i
1.38326 + 1.75687i
0.829861 + 2.07637i
−1.38326 1.75687i
−2.07637 + 0.829861i
−1.75687 + 1.38326i
0 0 0 −7.66648 13.2787i 0 0 0 0 0
521.2 0 0 0 −7.66648 13.2787i 0 0 0 0 0
521.3 0 0 0 −1.10680 1.91704i 0 0 0 0 0
521.4 0 0 0 −1.10680 1.91704i 0 0 0 0 0
521.5 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.6 0 0 0 1.10680 + 1.91704i 0 0 0 0 0
521.7 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
521.8 0 0 0 7.66648 + 13.2787i 0 0 0 0 0
1097.1 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.2 0 0 0 −7.66648 + 13.2787i 0 0 0 0 0
1097.3 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.4 0 0 0 −1.10680 + 1.91704i 0 0 0 0 0
1097.5 0 0 0 1.10680 1.91704i 0 0 0 0 0
1097.6 0 0 0 1.10680 1.91704i 0 0 0 0 0
1097.7 0 0 0 7.66648 13.2787i 0 0 0 0 0
1097.8 0 0 0 7.66648 13.2787i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1097.8
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 1764.4.t.a 16
3.b odd 2 1 inner 1764.4.t.a 16
7.b odd 2 1 inner 1764.4.t.a 16
7.c even 3 1 252.4.f.a 8
7.c even 3 1 inner 1764.4.t.a 16
7.d odd 6 1 252.4.f.a 8
7.d odd 6 1 inner 1764.4.t.a 16
21.c even 2 1 inner 1764.4.t.a 16
21.g even 6 1 252.4.f.a 8
21.g even 6 1 inner 1764.4.t.a 16
21.h odd 6 1 252.4.f.a 8
21.h odd 6 1 inner 1764.4.t.a 16
28.f even 6 1 1008.4.k.d 8
28.g odd 6 1 1008.4.k.d 8
84.j odd 6 1 1008.4.k.d 8
84.n even 6 1 1008.4.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.f.a 8 7.c even 3 1
252.4.f.a 8 7.d odd 6 1
252.4.f.a 8 21.g even 6 1
252.4.f.a 8 21.h odd 6 1
1008.4.k.d 8 28.f even 6 1
1008.4.k.d 8 28.g odd 6 1
1008.4.k.d 8 84.j odd 6 1
1008.4.k.d 8 84.n even 6 1
1764.4.t.a 16 1.a even 1 1 trivial
1764.4.t.a 16 3.b odd 2 1 inner
1764.4.t.a 16 7.b odd 2 1 inner
1764.4.t.a 16 7.c even 3 1 inner
1764.4.t.a 16 7.d odd 6 1 inner
1764.4.t.a 16 21.c even 2 1 inner
1764.4.t.a 16 21.g even 6 1 inner
1764.4.t.a 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 240 T_{5}^{6} + 56448 T_{5}^{4} + 276480 T_{5}^{2} + 1327104 \) acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 1327104 + 276480 T^{2} + 56448 T^{4} + 240 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 171396 - 414 T^{2} + T^{4} )^{4} \)
$13$ \( ( 3875328 + 5088 T^{2} + T^{4} )^{4} \)
$17$ \( ( 37903417344 + 3186653184 T^{2} + 267716736 T^{4} + 16368 T^{6} + T^{8} )^{2} \)
$19$ \( ( 2767192326144 - 36250730496 T^{2} + 473227776 T^{4} - 21792 T^{6} + T^{8} )^{2} \)
$23$ \( ( 8860231742470416 - 2070457172784 T^{2} + 389695212 T^{4} - 21996 T^{6} + T^{8} )^{2} \)
$29$ \( ( 1700572644 + 83124 T^{2} + T^{4} )^{4} \)
$31$ \( ( 85630228705050624 - 11124486438912 T^{2} + 1152589824 T^{4} - 38016 T^{6} + T^{8} )^{2} \)
$37$ \( ( 661106944 - 3291136 T + 42096 T^{2} + 128 T^{3} + T^{4} )^{4} \)
$41$ \( ( 22577275008 - 328560 T^{2} + T^{4} )^{4} \)
$43$ \( ( -20684 - 8 T + T^{2} )^{8} \)
$47$ \( ( \)\(17\!\cdots\!24\)\( + 16927197799514112 T^{2} + 124623489024 T^{4} + 407616 T^{6} + T^{8} )^{2} \)
$53$ \( ( \)\(12\!\cdots\!96\)\( - 3681637515747792 T^{2} + 92949818220 T^{4} - 323028 T^{6} + T^{8} )^{2} \)
$59$ \( ( 3932242550345170944 + 204073290694656 T^{2} + 8607891456 T^{4} + 102912 T^{6} + T^{8} )^{2} \)
$61$ \( ( \)\(49\!\cdots\!00\)\( - 15843852288000000 T^{2} + 482365440000 T^{4} - 710400 T^{6} + T^{8} )^{2} \)
$67$ \( ( 5186304256 - 42057344 T + 269040 T^{2} - 584 T^{3} + T^{4} )^{4} \)
$71$ \( ( 17756628516 + 331308 T^{2} + T^{4} )^{4} \)
$73$ \( ( \)\(28\!\cdots\!04\)\( - 24883615693651968 T^{2} + 161233519104 T^{4} - 463584 T^{6} + T^{8} )^{2} \)
$79$ \( ( 4768731136 - 30937088 T + 269760 T^{2} + 448 T^{3} + T^{4} )^{4} \)
$83$ \( ( 328711292928 - 1430592 T^{2} + T^{4} )^{4} \)
$89$ \( ( \)\(58\!\cdots\!84\)\( + 12166978337372160 T^{2} + 229115392128 T^{4} + 503280 T^{6} + T^{8} )^{2} \)
$97$ \( ( 149346427392 + 816096 T^{2} + T^{4} )^{4} \)
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