# Properties

 Label 768.4.d.n Level $768$ Weight $4$ Character orbit 768.d Analytic conductor $45.313$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(385,768)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 6 i q^{5} + 16 q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 6*i * q^5 + 16 * q^7 - 9 * q^9 $$q + 3 i q^{3} + 6 i q^{5} + 16 q^{7} - 9 q^{9} + 12 i q^{11} - 38 i q^{13} - 18 q^{15} - 126 q^{17} - 20 i q^{19} + 48 i q^{21} - 168 q^{23} + 89 q^{25} - 27 i q^{27} - 30 i q^{29} - 88 q^{31} - 36 q^{33} + 96 i q^{35} + 254 i q^{37} + 114 q^{39} - 42 q^{41} - 52 i q^{43} - 54 i q^{45} - 96 q^{47} - 87 q^{49} - 378 i q^{51} + 198 i q^{53} - 72 q^{55} + 60 q^{57} - 660 i q^{59} + 538 i q^{61} - 144 q^{63} + 228 q^{65} - 884 i q^{67} - 504 i q^{69} - 792 q^{71} - 218 q^{73} + 267 i q^{75} + 192 i q^{77} - 520 q^{79} + 81 q^{81} + 492 i q^{83} - 756 i q^{85} + 90 q^{87} - 810 q^{89} - 608 i q^{91} - 264 i q^{93} + 120 q^{95} + 1154 q^{97} - 108 i q^{99} +O(q^{100})$$ q + 3*i * q^3 + 6*i * q^5 + 16 * q^7 - 9 * q^9 + 12*i * q^11 - 38*i * q^13 - 18 * q^15 - 126 * q^17 - 20*i * q^19 + 48*i * q^21 - 168 * q^23 + 89 * q^25 - 27*i * q^27 - 30*i * q^29 - 88 * q^31 - 36 * q^33 + 96*i * q^35 + 254*i * q^37 + 114 * q^39 - 42 * q^41 - 52*i * q^43 - 54*i * q^45 - 96 * q^47 - 87 * q^49 - 378*i * q^51 + 198*i * q^53 - 72 * q^55 + 60 * q^57 - 660*i * q^59 + 538*i * q^61 - 144 * q^63 + 228 * q^65 - 884*i * q^67 - 504*i * q^69 - 792 * q^71 - 218 * q^73 + 267*i * q^75 + 192*i * q^77 - 520 * q^79 + 81 * q^81 + 492*i * q^83 - 756*i * q^85 + 90 * q^87 - 810 * q^89 - 608*i * q^91 - 264*i * q^93 + 120 * q^95 + 1154 * q^97 - 108*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q + 32 * q^7 - 18 * q^9 $$2 q + 32 q^{7} - 18 q^{9} - 36 q^{15} - 252 q^{17} - 336 q^{23} + 178 q^{25} - 176 q^{31} - 72 q^{33} + 228 q^{39} - 84 q^{41} - 192 q^{47} - 174 q^{49} - 144 q^{55} + 120 q^{57} - 288 q^{63} + 456 q^{65} - 1584 q^{71} - 436 q^{73} - 1040 q^{79} + 162 q^{81} + 180 q^{87} - 1620 q^{89} + 240 q^{95} + 2308 q^{97}+O(q^{100})$$ 2 * q + 32 * q^7 - 18 * q^9 - 36 * q^15 - 252 * q^17 - 336 * q^23 + 178 * q^25 - 176 * q^31 - 72 * q^33 + 228 * q^39 - 84 * q^41 - 192 * q^47 - 174 * q^49 - 144 * q^55 + 120 * q^57 - 288 * q^63 + 456 * q^65 - 1584 * q^71 - 436 * q^73 - 1040 * q^79 + 162 * q^81 + 180 * q^87 - 1620 * q^89 + 240 * q^95 + 2308 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 3.00000i 0 6.00000i 0 16.0000 0 −9.00000 0
385.2 0 3.00000i 0 6.00000i 0 16.0000 0 −9.00000 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 768.4.d.n 2
4.b odd 2 1 768.4.d.c 2
8.b even 2 1 inner 768.4.d.n 2
8.d odd 2 1 768.4.d.c 2
16.e even 4 1 6.4.a.a 1
16.e even 4 1 192.4.a.i 1
16.f odd 4 1 48.4.a.c 1
16.f odd 4 1 192.4.a.c 1
48.i odd 4 1 18.4.a.a 1
48.i odd 4 1 576.4.a.q 1
48.k even 4 1 144.4.a.c 1
48.k even 4 1 576.4.a.r 1
80.i odd 4 1 150.4.c.d 2
80.j even 4 1 1200.4.f.j 2
80.k odd 4 1 1200.4.a.b 1
80.q even 4 1 150.4.a.i 1
80.s even 4 1 1200.4.f.j 2
80.t odd 4 1 150.4.c.d 2
112.j even 4 1 2352.4.a.e 1
112.l odd 4 1 294.4.a.e 1
112.w even 12 2 294.4.e.h 2
112.x odd 12 2 294.4.e.g 2
144.w odd 12 2 162.4.c.c 2
144.x even 12 2 162.4.c.f 2
176.l odd 4 1 726.4.a.f 1
208.m odd 4 1 1014.4.b.d 2
208.p even 4 1 1014.4.a.g 1
208.r odd 4 1 1014.4.b.d 2
240.bb even 4 1 450.4.c.e 2
240.bf even 4 1 450.4.c.e 2
240.bm odd 4 1 450.4.a.h 1
272.r even 4 1 1734.4.a.d 1
304.j odd 4 1 2166.4.a.i 1
336.y even 4 1 882.4.a.n 1
336.bo even 12 2 882.4.g.f 2
336.bt odd 12 2 882.4.g.i 2
528.x even 4 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 16.e even 4 1
18.4.a.a 1 48.i odd 4 1
48.4.a.c 1 16.f odd 4 1
144.4.a.c 1 48.k even 4 1
150.4.a.i 1 80.q even 4 1
150.4.c.d 2 80.i odd 4 1
150.4.c.d 2 80.t odd 4 1
162.4.c.c 2 144.w odd 12 2
162.4.c.f 2 144.x even 12 2
192.4.a.c 1 16.f odd 4 1
192.4.a.i 1 16.e even 4 1
294.4.a.e 1 112.l odd 4 1
294.4.e.g 2 112.x odd 12 2
294.4.e.h 2 112.w even 12 2
450.4.a.h 1 240.bm odd 4 1
450.4.c.e 2 240.bb even 4 1
450.4.c.e 2 240.bf even 4 1
576.4.a.q 1 48.i odd 4 1
576.4.a.r 1 48.k even 4 1
726.4.a.f 1 176.l odd 4 1
768.4.d.c 2 4.b odd 2 1
768.4.d.c 2 8.d odd 2 1
768.4.d.n 2 1.a even 1 1 trivial
768.4.d.n 2 8.b even 2 1 inner
882.4.a.n 1 336.y even 4 1
882.4.g.f 2 336.bo even 12 2
882.4.g.i 2 336.bt odd 12 2
1014.4.a.g 1 208.p even 4 1
1014.4.b.d 2 208.m odd 4 1
1014.4.b.d 2 208.r odd 4 1
1200.4.a.b 1 80.k odd 4 1
1200.4.f.j 2 80.j even 4 1
1200.4.f.j 2 80.s even 4 1
1734.4.a.d 1 272.r even 4 1
2166.4.a.i 1 304.j odd 4 1
2178.4.a.e 1 528.x even 4 1
2352.4.a.e 1 112.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 36$$ T5^2 + 36 $$T_{7} - 16$$ T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 36$$
$7$ $$(T - 16)^{2}$$
$11$ $$T^{2} + 144$$
$13$ $$T^{2} + 1444$$
$17$ $$(T + 126)^{2}$$
$19$ $$T^{2} + 400$$
$23$ $$(T + 168)^{2}$$
$29$ $$T^{2} + 900$$
$31$ $$(T + 88)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T + 42)^{2}$$
$43$ $$T^{2} + 2704$$
$47$ $$(T + 96)^{2}$$
$53$ $$T^{2} + 39204$$
$59$ $$T^{2} + 435600$$
$61$ $$T^{2} + 289444$$
$67$ $$T^{2} + 781456$$
$71$ $$(T + 792)^{2}$$
$73$ $$(T + 218)^{2}$$
$79$ $$(T + 520)^{2}$$
$83$ $$T^{2} + 242064$$
$89$ $$(T + 810)^{2}$$
$97$ $$(T - 1154)^{2}$$