# Properties

 Label 192.7.e.e Level $192$ Weight $7$ Character orbit 192.e Analytic conductor $44.170$ Analytic rank $0$ 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] = [192,7,Mod(65,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (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: $$44.1703840550$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 12) 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 $$\beta = 12\sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 3) q^{3} + 6 \beta q^{5} + 242 q^{7} + (6 \beta - 711) q^{9}+O(q^{10})$$ q + (b + 3) * q^3 + 6*b * q^5 + 242 * q^7 + (6*b - 711) * q^9 $$q + (\beta + 3) q^{3} + 6 \beta q^{5} + 242 q^{7} + (6 \beta - 711) q^{9} - 66 \beta q^{11} - 2618 q^{13} + (18 \beta - 4320) q^{15} - 264 \beta q^{17} - 5786 q^{19} + (242 \beta + 726) q^{21} + 348 \beta q^{23} - 10295 q^{25} + ( - 693 \beta - 6453) q^{27} - 462 \beta q^{29} - 20446 q^{31} + ( - 198 \beta + 47520) q^{33} + 1452 \beta q^{35} + 46774 q^{37} + ( - 2618 \beta - 7854) q^{39} - 132 \beta q^{41} - 68618 q^{43} + ( - 4266 \beta - 25920) q^{45} + 792 \beta q^{47} - 59085 q^{49} + ( - 792 \beta + 190080) q^{51} - 6402 \beta q^{53} + 285120 q^{55} + ( - 5786 \beta - 17358) q^{57} + 5574 \beta q^{59} - 24794 q^{61} + (1452 \beta - 172062) q^{63} - 15708 \beta q^{65} + 84358 q^{67} + (1044 \beta - 250560) q^{69} + 12084 \beta q^{71} - 113806 q^{73} + ( - 10295 \beta - 30885) q^{75} - 15972 \beta q^{77} - 159742 q^{79} + ( - 8532 \beta + 479601) q^{81} - 19206 \beta q^{83} + 1140480 q^{85} + ( - 1386 \beta + 332640) q^{87} - 46812 \beta q^{89} - 633556 q^{91} + ( - 20446 \beta - 61338) q^{93} - 34716 \beta q^{95} + 899522 q^{97} + (46926 \beta + 285120) q^{99} +O(q^{100})$$ q + (b + 3) * q^3 + 6*b * q^5 + 242 * q^7 + (6*b - 711) * q^9 - 66*b * q^11 - 2618 * q^13 + (18*b - 4320) * q^15 - 264*b * q^17 - 5786 * q^19 + (242*b + 726) * q^21 + 348*b * q^23 - 10295 * q^25 + (-693*b - 6453) * q^27 - 462*b * q^29 - 20446 * q^31 + (-198*b + 47520) * q^33 + 1452*b * q^35 + 46774 * q^37 + (-2618*b - 7854) * q^39 - 132*b * q^41 - 68618 * q^43 + (-4266*b - 25920) * q^45 + 792*b * q^47 - 59085 * q^49 + (-792*b + 190080) * q^51 - 6402*b * q^53 + 285120 * q^55 + (-5786*b - 17358) * q^57 + 5574*b * q^59 - 24794 * q^61 + (1452*b - 172062) * q^63 - 15708*b * q^65 + 84358 * q^67 + (1044*b - 250560) * q^69 + 12084*b * q^71 - 113806 * q^73 + (-10295*b - 30885) * q^75 - 15972*b * q^77 - 159742 * q^79 + (-8532*b + 479601) * q^81 - 19206*b * q^83 + 1140480 * q^85 + (-1386*b + 332640) * q^87 - 46812*b * q^89 - 633556 * q^91 + (-20446*b - 61338) * q^93 - 34716*b * q^95 + 899522 * q^97 + (46926*b + 285120) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 484 q^{7} - 1422 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 484 * q^7 - 1422 * q^9 $$2 q + 6 q^{3} + 484 q^{7} - 1422 q^{9} - 5236 q^{13} - 8640 q^{15} - 11572 q^{19} + 1452 q^{21} - 20590 q^{25} - 12906 q^{27} - 40892 q^{31} + 95040 q^{33} + 93548 q^{37} - 15708 q^{39} - 137236 q^{43} - 51840 q^{45} - 118170 q^{49} + 380160 q^{51} + 570240 q^{55} - 34716 q^{57} - 49588 q^{61} - 344124 q^{63} + 168716 q^{67} - 501120 q^{69} - 227612 q^{73} - 61770 q^{75} - 319484 q^{79} + 959202 q^{81} + 2280960 q^{85} + 665280 q^{87} - 1267112 q^{91} - 122676 q^{93} + 1799044 q^{97} + 570240 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 484 * q^7 - 1422 * q^9 - 5236 * q^13 - 8640 * q^15 - 11572 * q^19 + 1452 * q^21 - 20590 * q^25 - 12906 * q^27 - 40892 * q^31 + 95040 * q^33 + 93548 * q^37 - 15708 * q^39 - 137236 * q^43 - 51840 * q^45 - 118170 * q^49 + 380160 * q^51 + 570240 * q^55 - 34716 * q^57 - 49588 * q^61 - 344124 * q^63 + 168716 * q^67 - 501120 * q^69 - 227612 * q^73 - 61770 * q^75 - 319484 * q^79 + 959202 * q^81 + 2280960 * q^85 + 665280 * q^87 - 1267112 * q^91 - 122676 * q^93 + 1799044 * q^97 + 570240 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 2.23607i 2.23607i
0 3.00000 26.8328i 0 160.997i 0 242.000 0 −711.000 160.997i 0
65.2 0 3.00000 + 26.8328i 0 160.997i 0 242.000 0 −711.000 + 160.997i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.7.e.e 2
3.b odd 2 1 inner 192.7.e.e 2
4.b odd 2 1 192.7.e.d 2
8.b even 2 1 12.7.c.a 2
8.d odd 2 1 48.7.e.c 2
12.b even 2 1 192.7.e.d 2
24.f even 2 1 48.7.e.c 2
24.h odd 2 1 12.7.c.a 2
40.f even 2 1 300.7.g.e 2
40.i odd 4 2 300.7.b.c 4
56.h odd 2 1 588.7.c.e 2
72.j odd 6 2 324.7.g.c 4
72.n even 6 2 324.7.g.c 4
120.i odd 2 1 300.7.g.e 2
120.w even 4 2 300.7.b.c 4
168.i even 2 1 588.7.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.7.c.a 2 8.b even 2 1
12.7.c.a 2 24.h odd 2 1
48.7.e.c 2 8.d odd 2 1
48.7.e.c 2 24.f even 2 1
192.7.e.d 2 4.b odd 2 1
192.7.e.d 2 12.b even 2 1
192.7.e.e 2 1.a even 1 1 trivial
192.7.e.e 2 3.b odd 2 1 inner
300.7.b.c 4 40.i odd 4 2
300.7.b.c 4 120.w even 4 2
300.7.g.e 2 40.f even 2 1
300.7.g.e 2 120.i odd 2 1
324.7.g.c 4 72.j odd 6 2
324.7.g.c 4 72.n even 6 2
588.7.c.e 2 56.h odd 2 1
588.7.c.e 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 25920$$ T5^2 + 25920 $$T_{7} - 242$$ T7 - 242

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6T + 729$$
$5$ $$T^{2} + 25920$$
$7$ $$(T - 242)^{2}$$
$11$ $$T^{2} + 3136320$$
$13$ $$(T + 2618)^{2}$$
$17$ $$T^{2} + 50181120$$
$19$ $$(T + 5786)^{2}$$
$23$ $$T^{2} + 87194880$$
$29$ $$T^{2} + 153679680$$
$31$ $$(T + 20446)^{2}$$
$37$ $$(T - 46774)^{2}$$
$41$ $$T^{2} + 12545280$$
$43$ $$(T + 68618)^{2}$$
$47$ $$T^{2} + 451630080$$
$53$ $$T^{2} + 29509634880$$
$59$ $$T^{2} + 22370022720$$
$61$ $$(T + 24794)^{2}$$
$67$ $$(T - 84358)^{2}$$
$71$ $$T^{2} + 105136600320$$
$73$ $$(T + 113806)^{2}$$
$79$ $$(T + 159742)^{2}$$
$83$ $$T^{2} + 265586713920$$
$89$ $$T^{2} + 1577781607680$$
$97$ $$(T - 899522)^{2}$$