# Properties

 Label 192.9.e.f Level $192$ Weight $9$ Character orbit 192.e Analytic conductor $78.217$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,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, 9, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$78.2166931317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 14$$ x^2 + 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 3) 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 $$\beta = 6\sqrt{-14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \beta + 45) q^{3} + 10 \beta q^{5} + 1750 q^{7} + ( - 270 \beta - 2511) q^{9}+O(q^{10})$$ q + (-3*b + 45) * q^3 + 10*b * q^5 + 1750 * q^7 + (-270*b - 2511) * q^9 $$q + ( - 3 \beta + 45) q^{3} + 10 \beta q^{5} + 1750 q^{7} + ( - 270 \beta - 2511) q^{9} + 310 \beta q^{11} - 25730 q^{13} + (450 \beta + 15120) q^{15} + 3336 \beta q^{17} + 18938 q^{19} + ( - 5250 \beta + 78750) q^{21} + 20956 \beta q^{23} + 340225 q^{25} + ( - 4617 \beta - 521235) q^{27} - 20530 \beta q^{29} + 351478 q^{31} + (13950 \beta + 468720) q^{33} + 17500 \beta q^{35} - 1335170 q^{37} + (77190 \beta - 1157850) q^{39} + 83540 \beta q^{41} - 3526150 q^{43} + ( - 25110 \beta + 1360800) q^{45} + 181784 \beta q^{47} - 2702301 q^{49} + (150120 \beta + 5044032) q^{51} + 294066 \beta q^{53} - 1562400 q^{55} + ( - 56814 \beta + 852210) q^{57} + 610910 \beta q^{59} - 753602 q^{61} + ( - 472500 \beta - 4394250) q^{63} - 257300 \beta q^{65} + 2268890 q^{67} + (943020 \beta + 31685472) q^{69} - 758220 \beta q^{71} + 27672770 q^{73} + ( - 1020675 \beta + 15310125) q^{75} + 542500 \beta q^{77} + 22980982 q^{79} + (1355940 \beta - 30436479) q^{81} - 2066606 \beta q^{83} - 16813440 q^{85} + ( - 923850 \beta - 31041360) q^{87} + 3234540 \beta q^{89} - 45027500 q^{91} + ( - 1054434 \beta + 15816510) q^{93} + 189380 \beta q^{95} + 147271010 q^{97} + ( - 778410 \beta + 42184800) q^{99} +O(q^{100})$$ q + (-3*b + 45) * q^3 + 10*b * q^5 + 1750 * q^7 + (-270*b - 2511) * q^9 + 310*b * q^11 - 25730 * q^13 + (450*b + 15120) * q^15 + 3336*b * q^17 + 18938 * q^19 + (-5250*b + 78750) * q^21 + 20956*b * q^23 + 340225 * q^25 + (-4617*b - 521235) * q^27 - 20530*b * q^29 + 351478 * q^31 + (13950*b + 468720) * q^33 + 17500*b * q^35 - 1335170 * q^37 + (77190*b - 1157850) * q^39 + 83540*b * q^41 - 3526150 * q^43 + (-25110*b + 1360800) * q^45 + 181784*b * q^47 - 2702301 * q^49 + (150120*b + 5044032) * q^51 + 294066*b * q^53 - 1562400 * q^55 + (-56814*b + 852210) * q^57 + 610910*b * q^59 - 753602 * q^61 + (-472500*b - 4394250) * q^63 - 257300*b * q^65 + 2268890 * q^67 + (943020*b + 31685472) * q^69 - 758220*b * q^71 + 27672770 * q^73 + (-1020675*b + 15310125) * q^75 + 542500*b * q^77 + 22980982 * q^79 + (1355940*b - 30436479) * q^81 - 2066606*b * q^83 - 16813440 * q^85 + (-923850*b - 31041360) * q^87 + 3234540*b * q^89 - 45027500 * q^91 + (-1054434*b + 15816510) * q^93 + 189380*b * q^95 + 147271010 * q^97 + (-778410*b + 42184800) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 90 q^{3} + 3500 q^{7} - 5022 q^{9}+O(q^{10})$$ 2 * q + 90 * q^3 + 3500 * q^7 - 5022 * q^9 $$2 q + 90 q^{3} + 3500 q^{7} - 5022 q^{9} - 51460 q^{13} + 30240 q^{15} + 37876 q^{19} + 157500 q^{21} + 680450 q^{25} - 1042470 q^{27} + 702956 q^{31} + 937440 q^{33} - 2670340 q^{37} - 2315700 q^{39} - 7052300 q^{43} + 2721600 q^{45} - 5404602 q^{49} + 10088064 q^{51} - 3124800 q^{55} + 1704420 q^{57} - 1507204 q^{61} - 8788500 q^{63} + 4537780 q^{67} + 63370944 q^{69} + 55345540 q^{73} + 30620250 q^{75} + 45961964 q^{79} - 60872958 q^{81} - 33626880 q^{85} - 62082720 q^{87} - 90055000 q^{91} + 31633020 q^{93} + 294542020 q^{97} + 84369600 q^{99}+O(q^{100})$$ 2 * q + 90 * q^3 + 3500 * q^7 - 5022 * q^9 - 51460 * q^13 + 30240 * q^15 + 37876 * q^19 + 157500 * q^21 + 680450 * q^25 - 1042470 * q^27 + 702956 * q^31 + 937440 * q^33 - 2670340 * q^37 - 2315700 * q^39 - 7052300 * q^43 + 2721600 * q^45 - 5404602 * q^49 + 10088064 * q^51 - 3124800 * q^55 + 1704420 * q^57 - 1507204 * q^61 - 8788500 * q^63 + 4537780 * q^67 + 63370944 * q^69 + 55345540 * q^73 + 30620250 * q^75 + 45961964 * q^79 - 60872958 * q^81 - 33626880 * q^85 - 62082720 * q^87 - 90055000 * q^91 + 31633020 * q^93 + 294542020 * q^97 + 84369600 * 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
 3.74166i − 3.74166i
0 45.0000 67.3498i 0 224.499i 0 1750.00 0 −2511.00 6061.48i 0
65.2 0 45.0000 + 67.3498i 0 224.499i 0 1750.00 0 −2511.00 + 6061.48i 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 192.9.e.f 2
3.b odd 2 1 inner 192.9.e.f 2
4.b odd 2 1 192.9.e.e 2
8.b even 2 1 48.9.e.b 2
8.d odd 2 1 3.9.b.a 2
12.b even 2 1 192.9.e.e 2
24.f even 2 1 3.9.b.a 2
24.h odd 2 1 48.9.e.b 2
40.e odd 2 1 75.9.c.c 2
40.k even 4 2 75.9.d.b 4
72.l even 6 2 81.9.d.d 4
72.p odd 6 2 81.9.d.d 4
120.m even 2 1 75.9.c.c 2
120.q odd 4 2 75.9.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 8.d odd 2 1
3.9.b.a 2 24.f even 2 1
48.9.e.b 2 8.b even 2 1
48.9.e.b 2 24.h odd 2 1
75.9.c.c 2 40.e odd 2 1
75.9.c.c 2 120.m even 2 1
75.9.d.b 4 40.k even 4 2
75.9.d.b 4 120.q odd 4 2
81.9.d.d 4 72.l even 6 2
81.9.d.d 4 72.p odd 6 2
192.9.e.e 2 4.b odd 2 1
192.9.e.e 2 12.b even 2 1
192.9.e.f 2 1.a even 1 1 trivial
192.9.e.f 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 50400$$ T5^2 + 50400 $$T_{7} - 1750$$ T7 - 1750

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 90T + 6561$$
$5$ $$T^{2} + 50400$$
$7$ $$(T - 1750)^{2}$$
$11$ $$T^{2} + 48434400$$
$13$ $$(T + 25730)^{2}$$
$17$ $$T^{2} + 5608963584$$
$19$ $$(T - 18938)^{2}$$
$23$ $$T^{2} + 221333583744$$
$29$ $$T^{2} + 212426373600$$
$31$ $$(T - 351478)^{2}$$
$37$ $$(T + 1335170)^{2}$$
$41$ $$T^{2} + 3517381526400$$
$43$ $$(T + 3526150)^{2}$$
$47$ $$T^{2} + 16654893018624$$
$53$ $$T^{2} + 43583305427424$$
$59$ $$T^{2} + 188098358162400$$
$61$ $$(T + 753602)^{2}$$
$67$ $$(T - 2268890)^{2}$$
$71$ $$T^{2} + 289748374473600$$
$73$ $$(T - 27672770)^{2}$$
$79$ $$(T - 22980982)^{2}$$
$83$ $$T^{2} + 21\!\cdots\!44$$
$89$ $$T^{2} + 52\!\cdots\!00$$
$97$ $$(T - 147271010)^{2}$$
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