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 $4$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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