# Properties

 Label 768.7.g.e Level $768$ Weight $7$ Character orbit 768.g Analytic conductor $176.682$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{6}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + 20 q^{5} + ( 12 \beta_{1} - \beta_{2} ) q^{7} -243 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + 20 q^{5} + ( 12 \beta_{1} - \beta_{2} ) q^{7} -243 q^{9} + ( 60 \beta_{1} + 4 \beta_{2} ) q^{11} + ( -1440 + \beta_{3} ) q^{13} + 20 \beta_{1} q^{15} + ( 558 + 4 \beta_{3} ) q^{17} + ( -12 \beta_{1} + 12 \beta_{2} ) q^{19} + ( -2916 - 3 \beta_{3} ) q^{21} + ( 456 \beta_{1} + 30 \beta_{2} ) q^{23} -15225 q^{25} -243 \beta_{1} q^{27} + ( 12652 - 18 \beta_{3} ) q^{29} + ( 492 \beta_{1} + 67 \beta_{2} ) q^{31} + ( -14580 + 12 \beta_{3} ) q^{33} + ( 240 \beta_{1} - 20 \beta_{2} ) q^{35} + ( -53064 - 5 \beta_{3} ) q^{37} + ( -1440 \beta_{1} - 81 \beta_{2} ) q^{39} + ( 12366 + 12 \beta_{3} ) q^{41} + ( 3996 \beta_{1} - 108 \beta_{2} ) q^{43} -4860 q^{45} + ( -5064 \beta_{1} - 282 \beta_{2} ) q^{47} + ( -34847 - 72 \beta_{3} ) q^{49} + ( 558 \beta_{1} - 324 \beta_{2} ) q^{51} + ( -64100 - 90 \beta_{3} ) q^{53} + ( 1200 \beta_{1} + 80 \beta_{2} ) q^{55} + ( 2916 + 36 \beta_{3} ) q^{57} + ( -7524 \beta_{1} + 240 \beta_{2} ) q^{59} + ( 20232 - 37 \beta_{3} ) q^{61} + ( -2916 \beta_{1} + 243 \beta_{2} ) q^{63} + ( -28800 + 20 \beta_{3} ) q^{65} + ( -2316 \beta_{1} + 1344 \beta_{2} ) q^{67} + ( -110808 + 90 \beta_{3} ) q^{69} + ( 15672 \beta_{1} - 606 \beta_{2} ) q^{71} + ( -9614 - 216 \beta_{3} ) q^{73} -15225 \beta_{1} q^{75} + ( 295056 - 36 \beta_{3} ) q^{77} + ( -35028 \beta_{1} + 75 \beta_{2} ) q^{79} + 59049 q^{81} + ( -8652 \beta_{1} - 1340 \beta_{2} ) q^{83} + ( 11160 + 80 \beta_{3} ) q^{85} + ( 12652 \beta_{1} + 1458 \beta_{2} ) q^{87} + ( -829854 + 40 \beta_{3} ) q^{89} + ( 21888 \beta_{1} + 468 \beta_{2} ) q^{91} + ( -119556 + 201 \beta_{3} ) q^{93} + ( -240 \beta_{1} + 240 \beta_{2} ) q^{95} + ( -616210 + 360 \beta_{3} ) q^{97} + ( -14580 \beta_{1} - 972 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 80 q^{5} - 972 q^{9} + O(q^{10})$$ $$4 q + 80 q^{5} - 972 q^{9} - 5760 q^{13} + 2232 q^{17} - 11664 q^{21} - 60900 q^{25} + 50608 q^{29} - 58320 q^{33} - 212256 q^{37} + 49464 q^{41} - 19440 q^{45} - 139388 q^{49} - 256400 q^{53} + 11664 q^{57} + 80928 q^{61} - 115200 q^{65} - 443232 q^{69} - 38456 q^{73} + 1180224 q^{77} + 236196 q^{81} + 44640 q^{85} - 3319416 q^{89} - 478224 q^{93} - 2464840 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-9 \nu^{3} + 45 \nu^{2} - 45 \nu + 54$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$-24 \nu^{3} + 24 \nu^{2} - 216 \nu - 48$$ $$\beta_{3}$$ $$=$$ $$($$$$-864 \nu^{3} - 5616$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 9 \beta_{2} + 48 \beta_{1} + 432$$$$)/1728$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} - 9 \beta_{2} + 432 \beta_{1} - 3888$$$$)/1728$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} - 5616$$$$)/864$$

## 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
 1.28078 − 2.21837i −0.780776 + 1.35234i −0.780776 − 1.35234i 1.28078 + 2.21837i
0 15.5885i 0 20.0000 0 529.850i 0 −243.000 0
511.2 0 15.5885i 0 20.0000 0 155.727i 0 −243.000 0
511.3 0 15.5885i 0 20.0000 0 155.727i 0 −243.000 0
511.4 0 15.5885i 0 20.0000 0 529.850i 0 −243.000 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 20$$ acting on $$S_{7}^{\mathrm{new}}(768, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 243 + T^{2} )^{2}$$
$5$ $$( -20 + T )^{4}$$
$7$ $$6808230144 + 304992 T^{2} + T^{4}$$
$11$ $$1010555709696 + 5509728 T^{2} + T^{4}$$
$13$ $$( -1099008 + 2880 T + T^{2} )^{2}$$
$17$ $$( -50450364 - 1116 T + T^{2} )^{2}$$
$19$ $$285122947021056 + 33911136 T^{2} + T^{4}$$
$23$ $$3049817413423104 + 312564096 T^{2} + T^{4}$$
$29$ $$( -867851888 - 25304 T + T^{2} )^{2}$$
$31$ $$219636481734441216 + 1172594016 T^{2} + T^{4}$$
$37$ $$( 2736472896 + 106128 T + T^{2} )^{2}$$
$41$ $$( -303937596 - 24732 T + T^{2} )^{2}$$
$43$ $$6298399499403757824 + 10501589088 T^{2} + T^{4}$$
$47$ $$9689976869755981824 + 31151806848 T^{2} + T^{4}$$
$53$ $$( -21589314800 + 128200 T + T^{2} )^{2}$$
$59$ $$48834094621847226624 + 41049200736 T^{2} + T^{4}$$
$61$ $$( -3933966528 - 40464 T + T^{2} )^{2}$$
$67$ $$44\!\cdots\!96$$$$+ 427110244704 T^{2} + T^{4}$$
$71$ $$27\!\cdots\!24$$$$+ 205670627712 T^{2} + T^{4}$$
$73$ $$( -147928769852 + 19228 T + T^{2} )^{2}$$
$79$ $$88\!\cdots\!44$$$$+ 597624861024 T^{2} + T^{4}$$
$83$ $$37\!\cdots\!84$$$$+ 458360917344 T^{2} + T^{4}$$
$89$ $$( 683581488516 + 1659708 T + T^{2} )^{2}$$
$97$ $$( -31455232700 + 1232420 T + T^{2} )^{2}$$