Properties

 Label 768.2.d.d Level $768$ Weight $2$ Character orbit 768.d Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 i q^{5} - q^{9}+O(q^{10})$$ q + i * q^3 + 2*i * q^5 - q^9 $$q + i q^{3} + 2 i q^{5} - q^{9} + 4 i q^{11} - 2 i q^{13} - 2 q^{15} + 2 q^{17} + 4 i q^{19} - 8 q^{23} + q^{25} - i q^{27} + 6 i q^{29} - 8 q^{31} - 4 q^{33} - 6 i q^{37} + 2 q^{39} + 6 q^{41} + 4 i q^{43} - 2 i q^{45} - 7 q^{49} + 2 i q^{51} + 2 i q^{53} - 8 q^{55} - 4 q^{57} + 4 i q^{59} - 2 i q^{61} + 4 q^{65} + 4 i q^{67} - 8 i q^{69} + 8 q^{71} - 10 q^{73} + i q^{75} + 8 q^{79} + q^{81} + 4 i q^{83} + 4 i q^{85} - 6 q^{87} + 6 q^{89} - 8 i q^{93} - 8 q^{95} + 2 q^{97} - 4 i q^{99} +O(q^{100})$$ q + i * q^3 + 2*i * q^5 - q^9 + 4*i * q^11 - 2*i * q^13 - 2 * q^15 + 2 * q^17 + 4*i * q^19 - 8 * q^23 + q^25 - i * q^27 + 6*i * q^29 - 8 * q^31 - 4 * q^33 - 6*i * q^37 + 2 * q^39 + 6 * q^41 + 4*i * q^43 - 2*i * q^45 - 7 * q^49 + 2*i * q^51 + 2*i * q^53 - 8 * q^55 - 4 * q^57 + 4*i * q^59 - 2*i * q^61 + 4 * q^65 + 4*i * q^67 - 8*i * q^69 + 8 * q^71 - 10 * q^73 + i * q^75 + 8 * q^79 + q^81 + 4*i * q^83 + 4*i * q^85 - 6 * q^87 + 6 * q^89 - 8*i * q^93 - 8 * q^95 + 2 * q^97 - 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{15} + 4 q^{17} - 16 q^{23} + 2 q^{25} - 16 q^{31} - 8 q^{33} + 4 q^{39} + 12 q^{41} - 14 q^{49} - 16 q^{55} - 8 q^{57} + 8 q^{65} + 16 q^{71} - 20 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} + 12 q^{89} - 16 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^15 + 4 * q^17 - 16 * q^23 + 2 * q^25 - 16 * q^31 - 8 * q^33 + 4 * q^39 + 12 * q^41 - 14 * q^49 - 16 * q^55 - 8 * q^57 + 8 * q^65 + 16 * q^71 - 20 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 + 12 * q^89 - 16 * q^95 + 4 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000i 0 0 0 −1.00000 0
385.2 0 1.00000i 0 2.00000i 0 0 0 −1.00000 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
8.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.d.d 2
3.b odd 2 1 2304.2.d.k 2
4.b odd 2 1 768.2.d.e 2
8.b even 2 1 inner 768.2.d.d 2
8.d odd 2 1 768.2.d.e 2
12.b even 2 1 2304.2.d.i 2
16.e even 4 1 48.2.a.a 1
16.e even 4 1 192.2.a.b 1
16.f odd 4 1 24.2.a.a 1
16.f odd 4 1 192.2.a.d 1
24.f even 2 1 2304.2.d.i 2
24.h odd 2 1 2304.2.d.k 2
48.i odd 4 1 144.2.a.b 1
48.i odd 4 1 576.2.a.b 1
48.k even 4 1 72.2.a.a 1
48.k even 4 1 576.2.a.d 1
80.i odd 4 1 1200.2.f.b 2
80.i odd 4 1 4800.2.f.bg 2
80.j even 4 1 600.2.f.e 2
80.j even 4 1 4800.2.f.d 2
80.k odd 4 1 600.2.a.h 1
80.k odd 4 1 4800.2.a.q 1
80.q even 4 1 1200.2.a.d 1
80.q even 4 1 4800.2.a.cc 1
80.s even 4 1 600.2.f.e 2
80.s even 4 1 4800.2.f.d 2
80.t odd 4 1 1200.2.f.b 2
80.t odd 4 1 4800.2.f.bg 2
112.j even 4 1 1176.2.a.i 1
112.j even 4 1 9408.2.a.h 1
112.l odd 4 1 2352.2.a.i 1
112.l odd 4 1 9408.2.a.cc 1
112.u odd 12 2 1176.2.q.i 2
112.v even 12 2 1176.2.q.a 2
112.w even 12 2 2352.2.q.l 2
112.x odd 12 2 2352.2.q.r 2
144.u even 12 2 648.2.i.b 2
144.v odd 12 2 648.2.i.g 2
144.w odd 12 2 1296.2.i.e 2
144.x even 12 2 1296.2.i.m 2
176.i even 4 1 2904.2.a.c 1
176.l odd 4 1 5808.2.a.s 1
208.l even 4 1 4056.2.c.e 2
208.o odd 4 1 4056.2.a.i 1
208.p even 4 1 8112.2.a.be 1
208.s even 4 1 4056.2.c.e 2
240.t even 4 1 1800.2.a.m 1
240.z odd 4 1 1800.2.f.c 2
240.bb even 4 1 3600.2.f.r 2
240.bd odd 4 1 1800.2.f.c 2
240.bf even 4 1 3600.2.f.r 2
240.bm odd 4 1 3600.2.a.v 1
272.k odd 4 1 6936.2.a.p 1
304.m even 4 1 8664.2.a.j 1
336.v odd 4 1 3528.2.a.d 1
336.y even 4 1 7056.2.a.q 1
336.br odd 12 2 3528.2.s.y 2
336.bu even 12 2 3528.2.s.j 2
528.s odd 4 1 8712.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 16.f odd 4 1
48.2.a.a 1 16.e even 4 1
72.2.a.a 1 48.k even 4 1
144.2.a.b 1 48.i odd 4 1
192.2.a.b 1 16.e even 4 1
192.2.a.d 1 16.f odd 4 1
576.2.a.b 1 48.i odd 4 1
576.2.a.d 1 48.k even 4 1
600.2.a.h 1 80.k odd 4 1
600.2.f.e 2 80.j even 4 1
600.2.f.e 2 80.s even 4 1
648.2.i.b 2 144.u even 12 2
648.2.i.g 2 144.v odd 12 2
768.2.d.d 2 1.a even 1 1 trivial
768.2.d.d 2 8.b even 2 1 inner
768.2.d.e 2 4.b odd 2 1
768.2.d.e 2 8.d odd 2 1
1176.2.a.i 1 112.j even 4 1
1176.2.q.a 2 112.v even 12 2
1176.2.q.i 2 112.u odd 12 2
1200.2.a.d 1 80.q even 4 1
1200.2.f.b 2 80.i odd 4 1
1200.2.f.b 2 80.t odd 4 1
1296.2.i.e 2 144.w odd 12 2
1296.2.i.m 2 144.x even 12 2
1800.2.a.m 1 240.t even 4 1
1800.2.f.c 2 240.z odd 4 1
1800.2.f.c 2 240.bd odd 4 1
2304.2.d.i 2 12.b even 2 1
2304.2.d.i 2 24.f even 2 1
2304.2.d.k 2 3.b odd 2 1
2304.2.d.k 2 24.h odd 2 1
2352.2.a.i 1 112.l odd 4 1
2352.2.q.l 2 112.w even 12 2
2352.2.q.r 2 112.x odd 12 2
2904.2.a.c 1 176.i even 4 1
3528.2.a.d 1 336.v odd 4 1
3528.2.s.j 2 336.bu even 12 2
3528.2.s.y 2 336.br odd 12 2
3600.2.a.v 1 240.bm odd 4 1
3600.2.f.r 2 240.bb even 4 1
3600.2.f.r 2 240.bf even 4 1
4056.2.a.i 1 208.o odd 4 1
4056.2.c.e 2 208.l even 4 1
4056.2.c.e 2 208.s even 4 1
4800.2.a.q 1 80.k odd 4 1
4800.2.a.cc 1 80.q even 4 1
4800.2.f.d 2 80.j even 4 1
4800.2.f.d 2 80.s even 4 1
4800.2.f.bg 2 80.i odd 4 1
4800.2.f.bg 2 80.t odd 4 1
5808.2.a.s 1 176.l odd 4 1
6936.2.a.p 1 272.k odd 4 1
7056.2.a.q 1 336.y even 4 1
8112.2.a.be 1 208.p even 4 1
8664.2.a.j 1 304.m even 4 1
8712.2.a.u 1 528.s odd 4 1
9408.2.a.h 1 112.j even 4 1
9408.2.a.cc 1 112.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7}$$ T7 $$T_{23} + 8$$ T23 + 8

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 4$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 8)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T - 2)^{2}$$