# Properties

 Label 192.13.e.f Level $192$ Weight $13$ Character orbit 192.e Analytic conductor $175.487$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,13,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, 13, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$13$$ 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: $$175.486812917$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4105x^{2} + 385000$$ x^4 + 4105*x^2 + 385000 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{7}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 75) q^{3} + ( - \beta_{2} + 3 \beta_1) q^{5} + ( - \beta_{3} + 28 \beta_1 + 3950) q^{7} + ( - 12 \beta_{3} - 27 \beta_{2} + \cdots + 24201) q^{9}+O(q^{10})$$ q + (-b1 - 75) * q^3 + (-b2 + 3*b1) * q^5 + (-b3 + 28*b1 + 3950) * q^7 + (-12*b3 - 27*b2 + 69*b1 + 24201) * q^9 $$q + ( - \beta_1 - 75) q^{3} + ( - \beta_{2} + 3 \beta_1) q^{5} + ( - \beta_{3} + 28 \beta_1 + 3950) q^{7} + ( - 12 \beta_{3} - 27 \beta_{2} + \cdots + 24201) q^{9}+ \cdots + (37353663 \beta_{3} + \cdots - 318016504080) q^{99}+O(q^{100})$$ q + (-b1 - 75) * q^3 + (-b2 + 3*b1) * q^5 + (-b3 + 28*b1 + 3950) * q^7 + (-12*b3 - 27*b2 + 69*b1 + 24201) * q^9 + (63*b3 - 52*b2 + 1794*b1) * q^11 + (236*b3 - 6608*b1 - 858050) * q^13 + (-585*b3 + 324*b2 - 162*b1 + 1653480) * q^15 + (-1116*b3 - 1000*b2 - 26016*b1) * q^17 + (1175*b3 - 32900*b1 + 512506) * q^19 + (324*b3 + 729*b2 - 1763*b1 - 14994834) * q^21 + (3438*b3 + 6800*b2 + 68988*b1) * q^23 + (7740*b3 - 216720*b1 - 109362815) * q^25 + (-16083*b3 + 8100*b2 - 4419*b1 - 127089675) * q^27 + (-15336*b3 + 20669*b2 - 460743*b1) * q^29 + (20475*b3 - 573300*b1 - 629752066) * q^31 + (-10008*b3 + 62775*b2 - 125793*b1 + 948796200) * q^33 + (15570*b3 - 17396*b2 + 457008*b1) * q^35 + (105268*b3 - 2947504*b1 + 1866677950) * q^37 + (-76464*b3 - 172044*b2 + 341918*b1 + 3533219574) * q^39 + (-211824*b3 + 30046*b2 - 5597562*b1) * q^41 + (8991*b3 - 251748*b1 + 6529982650) * q^43 + (192240*b3 - 98901*b2 - 415017*b1 - 8969181840) * q^45 + (339588*b3 + 121800*b2 + 8463888*b1) * q^47 + (-7900*b3 + 221200*b1 - 13031961165) * q^49 + (-946584*b3 - 489564*b2 + 1917108*b1 - 13630675680) * q^51 + (-88992*b3 + 887675*b2 - 4976817*b1) * q^53 + (1411740*b3 - 39528720*b1 - 24007317840) * q^55 + (-380700*b3 - 856575*b2 - 3082231*b1 + 17232398250) * q^57 + (-1320993*b3 - 817328*b2 - 31893834*b1) * q^59 + (1731300*b3 - 48476400*b1 + 2326808926) * q^61 + (435441*b3 - 216000*b2 + 14461320*b1 + 4505169150) * q^63 + (-3674520*b3 + 4031306*b2 - 107631438*b1) * q^65 + (466851*b3 - 13071828*b1 - 22263896150) * q^67 + (5091912*b3 + 303102*b2 - 5303394*b1 + 35841396240) * q^69 + (-483822*b3 - 10716712*b2 + 19570764*b1) * q^71 + (-4360368*b3 + 122090304*b1 + 116035618850) * q^73 + (-2507760*b3 - 5642460*b2 + 92435435*b1 + 121969251285) * q^75 + (211392*b3 - 3384650*b2 + 15650142*b1) * q^77 + (6377275*b3 - 178563700*b1 + 141755506622) * q^79 + (4784076*b3 - 2521854*b2 + 160992738*b1 - 232262879679) * q^81 + (-10164051*b3 - 19092900*b2 - 206986626*b1) * q^83 + (-10138320*b3 + 283872960*b1 - 253858121280) * q^85 + (7122501*b3 - 17876700*b2 + 31919346*b1 - 244209659400) * q^87 + (8226324*b3 - 26619346*b2 + 293742462*b1) * q^89 + (1790250*b3 - 50127000*b1 - 190708051996) * q^91 + (-6633900*b3 - 14926275*b2 + 584973241*b1 + 348184912350) * q^93 + (-18294750*b3 + 15286544*b2 - 521523132*b1) * q^95 + (-5141996*b3 + 143975888*b1 + 215739631250) * q^97 + (37353663*b3 - 18920952*b2 - 918045306*b1 - 318016504080) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 300 q^{3} + 15800 q^{7} + 96804 q^{9}+O(q^{10})$$ 4 * q - 300 * q^3 + 15800 * q^7 + 96804 * q^9 $$4 q - 300 q^{3} + 15800 q^{7} + 96804 q^{9} - 3432200 q^{13} + 6613920 q^{15} + 2050024 q^{19} - 59979336 q^{21} - 437451260 q^{25} - 508358700 q^{27} - 2519008264 q^{31} + 3795184800 q^{33} + 7466711800 q^{37} + 14132878296 q^{39} + 26119930600 q^{43} - 35876727360 q^{45} - 52127844660 q^{49} - 54522702720 q^{51} - 96029271360 q^{55} + 68929593000 q^{57} + 9307235704 q^{61} + 18020676600 q^{63} - 89055584600 q^{67} + 143365584960 q^{69} + 464142475400 q^{73} + 487877005140 q^{75} + 567022026488 q^{79} - 929051518716 q^{81} - 1015432485120 q^{85} - 976838637600 q^{87} - 762832207984 q^{91} + 1392739649400 q^{93} + 862958525000 q^{97} - 1272066016320 q^{99}+O(q^{100})$$ 4 * q - 300 * q^3 + 15800 * q^7 + 96804 * q^9 - 3432200 * q^13 + 6613920 * q^15 + 2050024 * q^19 - 59979336 * q^21 - 437451260 * q^25 - 508358700 * q^27 - 2519008264 * q^31 + 3795184800 * q^33 + 7466711800 * q^37 + 14132878296 * q^39 + 26119930600 * q^43 - 35876727360 * q^45 - 52127844660 * q^49 - 54522702720 * q^51 - 96029271360 * q^55 + 68929593000 * q^57 + 9307235704 * q^61 + 18020676600 * q^63 - 89055584600 * q^67 + 143365584960 * q^69 + 464142475400 * q^73 + 487877005140 * q^75 + 567022026488 * q^79 - 929051518716 * q^81 - 1015432485120 * q^85 - 976838637600 * q^87 - 762832207984 * q^91 + 1392739649400 * q^93 + 862958525000 * q^97 - 1272066016320 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4105x^{2} + 385000$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 20\nu^{2} + 4505\nu - 41050 ) / 75$$ (v^3 - 20*v^2 + 4505*v - 41050) / 75 $$\beta_{2}$$ $$=$$ $$( 11\nu^{3} - 20\nu^{2} + 35155\nu - 41050 ) / 25$$ (11*v^3 - 20*v^2 + 35155*v - 41050) / 25 $$\beta_{3}$$ $$=$$ $$( 28\nu^{3} + 520\nu^{2} + 126140\nu + 1067300 ) / 75$$ (28*v^3 + 520*v^2 + 126140*v + 1067300) / 75
 $$\nu$$ $$=$$ $$( 5\beta_{3} - 9\beta_{2} + 157\beta_1 ) / 5184$$ (5*b3 - 9*b2 + 157*b1) / 5184 $$\nu^{2}$$ $$=$$ $$( 5\beta_{3} - 140\beta _1 - 147780 ) / 72$$ (5*b3 - 140*b1 - 147780) / 72 $$\nu^{3}$$ $$=$$ $$( -15325\beta_{3} + 40545\beta_{2} - 520085\beta_1 ) / 5184$$ (-15325*b3 + 40545*b2 - 520085*b1) / 5184

## 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
 63.3164i − 63.3164i 9.79973i − 9.79973i
0 −596.724 418.762i 0 23907.4i 0 32123.1 0 180718. + 499770.i 0
65.2 0 −596.724 + 418.762i 0 23907.4i 0 32123.1 0 180718. 499770.i 0
65.3 0 446.724 576.089i 0 11638.0i 0 −24223.1 0 −132316. 514706.i 0
65.4 0 446.724 + 576.089i 0 11638.0i 0 −24223.1 0 −132316. + 514706.i 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 192.13.e.f 4
3.b odd 2 1 inner 192.13.e.f 4
4.b odd 2 1 192.13.e.g 4
8.b even 2 1 12.13.c.a 4
8.d odd 2 1 48.13.e.d 4
12.b even 2 1 192.13.e.g 4
24.f even 2 1 48.13.e.d 4
24.h odd 2 1 12.13.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.13.c.a 4 8.b even 2 1
12.13.c.a 4 24.h odd 2 1
48.13.e.d 4 8.d odd 2 1
48.13.e.d 4 24.f even 2 1
192.13.e.f 4 1.a even 1 1 trivial
192.13.e.f 4 3.b odd 2 1 inner
192.13.e.g 4 4.b odd 2 1
192.13.e.g 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{13}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{4} + 707006880T_{5}^{2} + 77414609986560000$$ T5^4 + 707006880*T5^2 + 77414609986560000 $$T_{7}^{2} - 7900T_{7} - 778121036$$ T7^2 - 7900*T7 - 778121036

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 282429536481$$
$5$ $$T^{4} + \cdots + 77\!\cdots\!00$$
$7$ $$(T^{2} - 7900 T - 778121036)^{2}$$
$11$ $$T^{4} + \cdots + 13\!\cdots\!00$$
$13$ $$(T^{2} + \cdots - 43470976258556)^{2}$$
$17$ $$T^{4} + \cdots + 37\!\cdots\!00$$
$19$ $$(T^{2} + \cdots - 10\!\cdots\!64)^{2}$$
$23$ $$T^{4} + \cdots + 24\!\cdots\!00$$
$29$ $$T^{4} + \cdots + 39\!\cdots\!00$$
$31$ $$(T^{2} + \cdots + 63\!\cdots\!56)^{2}$$
$37$ $$(T^{2} + \cdots - 53\!\cdots\!64)^{2}$$
$41$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$43$ $$(T^{2} + \cdots + 42\!\cdots\!84)^{2}$$
$47$ $$T^{4} + \cdots + 32\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 30\!\cdots\!00$$
$59$ $$T^{4} + \cdots + 27\!\cdots\!00$$
$61$ $$(T^{2} + \cdots - 23\!\cdots\!24)^{2}$$
$67$ $$(T^{2} + \cdots + 32\!\cdots\!64)^{2}$$
$71$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$73$ $$(T^{2} + \cdots - 16\!\cdots\!64)^{2}$$
$79$ $$(T^{2} + \cdots - 12\!\cdots\!16)^{2}$$
$83$ $$T^{4} + \cdots + 14\!\cdots\!00$$
$89$ $$T^{4} + \cdots + 19\!\cdots\!00$$
$97$ $$(T^{2} + \cdots + 25\!\cdots\!24)^{2}$$