Properties

Label 768.7.g.g
Level $768$
Weight $7$
Character orbit 768.g
Analytic conductor $176.682$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(176.681536220\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 53 x^{6} - 2 x^{5} + 2532 x^{4} - 772 x^{3} - 31349 x^{2} - 33880 x + 366025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} -243 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} -243 q^{9} + ( -17 \beta_{1} - \beta_{4} ) q^{11} -\beta_{5} q^{13} + \beta_{7} q^{15} + ( -818 + \beta_{6} ) q^{17} + ( 195 \beta_{1} - 5 \beta_{4} ) q^{19} + ( -3 \beta_{2} + 3 \beta_{5} ) q^{21} + ( -2 \beta_{3} - 4 \beta_{7} ) q^{23} + ( 7079 + 2 \beta_{6} ) q^{25} + 243 \beta_{1} q^{27} + ( -47 \beta_{2} - 14 \beta_{5} ) q^{29} + ( -61 \beta_{3} + 20 \beta_{7} ) q^{31} + ( -4212 - 3 \beta_{6} ) q^{33} + ( 325 \beta_{1} + 33 \beta_{4} ) q^{35} + ( -110 \beta_{2} - 27 \beta_{5} ) q^{37} + ( -81 \beta_{3} + \beta_{7} ) q^{39} + ( 62446 - \beta_{6} ) q^{41} + ( 529 \beta_{1} + 25 \beta_{4} ) q^{43} + 243 \beta_{2} q^{45} + ( 198 \beta_{3} + 32 \beta_{7} ) q^{47} + ( -51839 + 12 \beta_{6} ) q^{49} + ( 845 \beta_{1} - 81 \beta_{4} ) q^{51} + ( 949 \beta_{2} - 70 \beta_{5} ) q^{53} + ( 400 \beta_{3} + 76 \beta_{7} ) q^{55} + ( 46980 - 15 \beta_{6} ) q^{57} + ( -10620 \beta_{1} - 32 \beta_{4} ) q^{59} + ( 462 \beta_{2} + 5 \beta_{5} ) q^{61} + 243 \beta_{3} q^{63} + ( -4512 - 31 \beta_{6} ) q^{65} + ( 10568 \beta_{1} + 140 \beta_{4} ) q^{67} + ( -978 \beta_{2} + 6 \beta_{5} ) q^{69} + ( -94 \beta_{3} + 228 \beta_{7} ) q^{71} + ( 245330 + 20 \beta_{6} ) q^{73} + ( -7025 \beta_{1} - 162 \beta_{4} ) q^{75} + ( -3284 \beta_{2} - 124 \beta_{5} ) q^{77} + ( -1525 \beta_{3} - 4 \beta_{7} ) q^{79} + 59049 q^{81} + ( 13683 \beta_{1} - 309 \beta_{4} ) q^{83} + ( -4334 \beta_{2} + 400 \beta_{5} ) q^{85} + ( -1134 \beta_{3} + 61 \beta_{7} ) q^{87} + ( 211874 - 38 \beta_{6} ) q^{89} + ( -56279 \beta_{1} + 357 \beta_{4} ) q^{91} + ( 4677 \beta_{2} + 183 \beta_{5} ) q^{93} + ( 2000 \beta_{3} + 100 \beta_{7} ) q^{95} + ( 954094 - 10 \beta_{6} ) q^{97} + ( 4131 \beta_{1} + 243 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1944 q^{9} + O(q^{10}) \) \( 8 q - 1944 q^{9} - 6544 q^{17} + 56632 q^{25} - 33696 q^{33} + 499568 q^{41} - 414712 q^{49} + 375840 q^{57} - 36096 q^{65} + 1962640 q^{73} + 472392 q^{81} + 1694992 q^{89} + 7632752 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 53 x^{6} - 2 x^{5} + 2532 x^{4} - 772 x^{3} - 31349 x^{2} - 33880 x + 366025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 828747 \nu^{7} + 5475258 \nu^{6} - 46400868 \nu^{5} - 318001302 \nu^{4} + 1561303692 \nu^{3} + 16532702568 \nu^{2} - 13895005095 \nu - 148555417725 \)\()/ 9239308925 \)
\(\beta_{2}\)\(=\)\((\)\( -26514 \nu^{7} + 579378 \nu^{6} + 11306672 \nu^{5} - 31230312 \nu^{4} - 392579888 \nu^{3} + 356343028 \nu^{2} + 9273241226 \nu - 974690090 \)\()/ 263980255 \)
\(\beta_{3}\)\(=\)\((\)\( -6425154 \nu^{7} + 3940254 \nu^{6} + 284961216 \nu^{5} + 483802824 \nu^{4} - 12869361504 \nu^{3} - 10886465316 \nu^{2} + 33621027330 \nu + 296213011050 \)\()/ 839937175 \)
\(\beta_{4}\)\(=\)\((\)\(-19492379 \nu^{7} - 175563642 \nu^{6} + 1247954852 \nu^{5} + 11629186678 \nu^{4} - 37349169868 \nu^{3} - 397943114152 \nu^{2} + 245471527351 \nu + 3833648818285\)\()/ 1847861785 \)
\(\beta_{5}\)\(=\)\((\)\(-22028796 \nu^{7} + 66004572 \nu^{6} + 1655284448 \nu^{5} - 1485058128 \nu^{4} - 77436405152 \nu^{3} + 39175429432 \nu^{2} + 1788467062604 \nu + 427089143140\)\()/ 1847861785 \)
\(\beta_{6}\)\(=\)\((\)\( -49248 \nu^{7} - 678816 \nu^{6} + 2184192 \nu^{5} + 3708288 \nu^{4} - 4436352 \nu^{3} - 83443392 \nu^{2} + 257700960 \nu - 24850917216 \)\()/3054317\)
\(\beta_{7}\)\(=\)\((\)\( 1068066 \nu^{7} - 986094 \nu^{6} - 47369664 \nu^{5} - 80423496 \nu^{4} + 1635589152 \nu^{3} + 1809678564 \nu^{2} - 5588889570 \nu - 39175581258 \)\()/33597487\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{5} - 9 \beta_{3} - 30 \beta_{2} - 48 \beta_{1} + 432\)\()/1728\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} + 27 \beta_{4} + 18 \beta_{3} - 60 \beta_{2} + 5271 \beta_{1} + 47520\)\()/3456\)
\(\nu^{3}\)\(=\)\((\)\(-212 \beta_{7} + 9 \beta_{6} - 900 \beta_{3} + 143424\)\()/3456\)
\(\nu^{4}\)\(=\)\((\)\(-107 \beta_{7} - 81 \beta_{6} + 153 \beta_{5} + 729 \beta_{4} - 459 \beta_{3} - 3042 \beta_{2} + 86877 \beta_{1} - 784080\)\()/1728\)
\(\nu^{5}\)\(=\)\((\)\(-9032 \beta_{7} + 825 \beta_{6} - 9288 \beta_{5} + 7425 \beta_{4} - 27864 \beta_{3} + 253152 \beta_{2} + 895317 \beta_{1} + 8080128\)\()/6912\)
\(\nu^{6}\)\(=\)\((\)\(-7042 \beta_{7} - 3423 \beta_{6} - 22050 \beta_{3} - 28285632\)\()/864\)
\(\nu^{7}\)\(=\)\((\)\(88549 \beta_{7} - 12705 \beta_{6} - 80847 \beta_{5} + 114345 \beta_{4} + 242541 \beta_{3} + 2471670 \beta_{2} + 11752077 \beta_{1} - 106111728\)\()/1728\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
−5.30399 3.63961i
3.53301 + 1.46244i
−3.03301 2.32846i
5.80399 + 2.77359i
−5.30399 + 3.63961i
3.53301 1.46244i
−3.03301 + 2.32846i
5.80399 2.77359i
0 15.5885i 0 −195.235 0 277.510i 0 −243.000 0
511.2 0 15.5885i 0 −85.3891 0 511.824i 0 −243.000 0
511.3 0 15.5885i 0 85.3891 0 511.824i 0 −243.000 0
511.4 0 15.5885i 0 195.235 0 277.510i 0 −243.000 0
511.5 0 15.5885i 0 −195.235 0 277.510i 0 −243.000 0
511.6 0 15.5885i 0 −85.3891 0 511.824i 0 −243.000 0
511.7 0 15.5885i 0 85.3891 0 511.824i 0 −243.000 0
511.8 0 15.5885i 0 195.235 0 277.510i 0 −243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.7.g.g 8
4.b odd 2 1 inner 768.7.g.g 8
8.b even 2 1 inner 768.7.g.g 8
8.d odd 2 1 inner 768.7.g.g 8
16.e even 4 2 384.7.b.c 8
16.f odd 4 2 384.7.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.c 8 16.e even 4 2
384.7.b.c 8 16.f odd 4 2
768.7.g.g 8 1.a even 1 1 trivial
768.7.g.g 8 4.b odd 2 1 inner
768.7.g.g 8 8.b even 2 1 inner
768.7.g.g 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 45408 T_{5}^{2} + 277920000 \) acting on \(S_{7}^{\mathrm{new}}(768, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 243 + T^{2} )^{4} \)
$5$ \( ( 277920000 - 45408 T^{2} + T^{4} )^{2} \)
$7$ \( ( 20174323968 + 338976 T^{2} + T^{4} )^{2} \)
$11$ \( ( 4522189383936 + 4545120 T^{2} + T^{4} )^{2} \)
$13$ \( ( 11711515004928 - 9088896 T^{2} + T^{4} )^{2} \)
$17$ \( ( -58718780 + 1636 T + T^{2} )^{4} \)
$19$ \( ( 2107360836000000 + 128143200 T^{2} + T^{4} )^{2} \)
$23$ \( ( 2996687502839808 + 180514944 T^{2} + T^{4} )^{2} \)
$29$ \( ( 123449102940268800 - 1869854304 T^{2} + T^{4} )^{2} \)
$31$ \( ( 6459874959912599808 + 5276545056 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1035562157965836288 - 7121639424 T^{2} + T^{4} )^{2} \)
$41$ \( ( 3840115012 - 124892 T + T^{2} )^{4} \)
$43$ \( ( 1701874998525913344 + 2889761376 T^{2} + T^{4} )^{2} \)
$47$ \( ( 80088648191955505152 + 26657465472 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(91\!\cdots\!00\)\( - 86629009248 T^{2} + T^{4} )^{2} \)
$59$ \( ( \)\(63\!\cdots\!00\)\( + 59428064352 T^{2} + T^{4} )^{2} \)
$61$ \( ( 19934164088443699200 - 9877596672 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(24\!\cdots\!84\)\( + 140980616544 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(55\!\cdots\!92\)\( + 569594613888 T^{2} + T^{4} )^{2} \)
$73$ \( ( 36431647300 - 490660 T + T^{2} )^{4} \)
$79$ \( ( \)\(11\!\cdots\!00\)\( + 790499817504 T^{2} + T^{4} )^{2} \)
$83$ \( ( \)\(27\!\cdots\!44\)\( + 509657219424 T^{2} + T^{4} )^{2} \)
$89$ \( ( -40865541500 - 423748 T + T^{2} )^{4} \)
$97$ \( ( 904356570436 - 1908188 T + T^{2} )^{4} \)
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