# Properties

 Label 192.9.e.j Level $192$ Weight $9$ Character orbit 192.e Analytic conductor $78.217$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633$$ x^8 - 4*x^7 - 78*x^6 + 144*x^5 + 2079*x^4 + 936*x^3 - 658*x^2 + 2884*x + 30633 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{42}\cdot 3^{10}$$ Twist minimal: no (minimal twist has level 24) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_1 - 198) q^{7} + ( - \beta_{5} + 2 \beta_{3} + \cdots + 41) q^{9}+O(q^{10})$$ q + (-b2 + 7) * q^3 + (b3 - b2) * q^5 + (-b1 - 198) * q^7 + (-b5 + 2*b3 - 8*b2 - 2*b1 + 41) * q^9 $$q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_1 - 198) q^{7} + ( - \beta_{5} + 2 \beta_{3} + \cdots + 41) q^{9}+ \cdots + ( - 17025 \beta_{7} + 31716 \beta_{6} + \cdots - 46212688) q^{99}+O(q^{100})$$ q + (-b2 + 7) * q^3 + (b3 - b2) * q^5 + (-b1 - 198) * q^7 + (-b5 + 2*b3 - 8*b2 - 2*b1 + 41) * q^9 + (-b7 - 2*b5 + b4 + 7*b3 + b2) * q^11 + (-2*b7 - b6 + 4*b5 + b4 + b3 - 52*b2 - 8*b1 - 3154) * q^13 + (6*b7 + 4*b6 + 6*b5 - b4 + 45*b3 - 22*b2 - 13*b1 - 4792) * q^15 + (10*b7 + 7*b6 - 10*b5 + 5*b4 - 81*b3 - 270*b2) * q^17 + (-25*b7 + 4*b6 - 16*b5 - 4*b4 - 4*b3 - 683*b2 - 10*b1 + 19742) * q^19 + (-42*b7 + 39*b6 + 6*b5 - 3*b4 - 138*b3 + 231*b2 - 36*b1 - 3810) * q^21 + (66*b7 - 68*b6 - 112*b3 - 1806*b2) * q^23 + (98*b7 - 5*b6 + 20*b5 + 5*b4 + 5*b3 + 2656*b2 - 28*b1 - 72463) * q^25 + (-141*b7 - 180*b6 + 34*b5 - 9*b4 - 95*b3 - 196*b2 - 130*b1 - 34505) * q^27 + (-212*b7 - 294*b6 - 76*b5 + 38*b4 - 203*b3 + 4617*b2) * q^29 + (222*b7 + 36*b6 - 144*b5 - 36*b4 - 36*b3 + 5922*b2 - 127*b1 - 100694) * q^31 + (270*b7 - 559*b6 + 99*b5 - 29*b4 - 597*b3 - 182*b2 - 230*b1 + 12856) * q^33 + (-308*b7 + 792*b6 + 110*b5 - 55*b4 - 353*b3 + 11298*b2) * q^35 + (-390*b7 + 9*b6 - 36*b5 - 9*b4 - 9*b3 - 10548*b2 + 304*b1 + 498126) * q^37 + (312*b7 + 1320*b6 - 42*b5 - 33*b4 - 2355*b3 + 4516*b2 - 450*b1 + 287138) * q^39 + (160*b7 + 1596*b6 - 112*b5 + 56*b4 + 1498*b3 - 3690*b2) * q^41 + (-253*b7 + 100*b6 - 400*b5 - 100*b4 - 100*b3 - 7031*b2 - 950*b1 + 870334) * q^43 + (-228*b7 + 2358*b6 + 250*b5 - 90*b4 + 877*b3 + 4061*b2 - 724*b1 - 1083824) * q^45 + (-96*b7 - 2600*b6 + 420*b5 - 210*b4 + 3106*b3 - 1724*b2) * q^47 + (-770*b7 + 77*b6 - 308*b5 - 77*b4 - 77*b3 - 20944*b2 + 2220*b1 - 735565) * q^49 + (654*b7 - 3224*b6 - 606*b5 - 85*b4 - 7227*b3 - 1528*b2 - 340*b1 - 1948768) * q^51 + (576*b7 - 3416*b6 + 480*b5 - 240*b4 + 4565*b3 - 22389*b2) * q^53 + (-1578*b7 - 12*b6 + 48*b5 + 12*b4 + 12*b3 - 42582*b2 - 2902*b1 - 3387664) * q^55 + (-3012*b7 - 3642*b6 - 471*b5 + 6*b4 + 5406*b3 - 18458*b2 - 1494*b1 + 4594586) * q^57 + (2361*b7 + 3568*b6 - 24*b5 + 12*b4 + 14868*b3 - 71707*b2) * q^59 + (1250*b7 - 11*b6 + 44*b5 + 11*b4 + 11*b3 + 33772*b2 + 4640*b1 + 6447950) * q^61 + (-3024*b7 + 2808*b6 - 744*b5 - 216*b4 - 7584*b3 + 17052*b2 + 1617*b1 + 8719914) * q^63 + (-5268*b7 + 2182*b6 + 1380*b5 - 690*b4 - 6720*b3 + 166430*b2) * q^65 + (3365*b7 - 488*b6 + 1952*b5 + 488*b4 + 488*b3 + 91831*b2 - 392*b1 + 7275086) * q^67 + (-672*b7 - 788*b6 - 2454*b5 + 656*b4 - 3108*b3 - 12472*b2 - 3196*b1 - 11778256) * q^69 + (-3102*b7 + 2020*b6 - 2364*b5 + 1182*b4 + 18306*b3 + 47030*b2) * q^71 + (-6832*b7 - 524*b6 + 2096*b5 + 524*b4 + 524*b3 - 183416*b2 - 1656*b1 - 14606846) * q^73 + (2064*b7 + 6132*b6 + 2634*b5 - 129*b4 - 15951*b3 + 61381*b2 + 4014*b1 - 17897737) * q^75 + (-5936*b7 + 8336*b6 - 880*b5 + 440*b4 - 29166*b3 + 197750*b2) * q^77 + (1086*b7 - 516*b6 + 2064*b5 + 516*b4 + 516*b3 + 30354*b2 + 17329*b1 - 21556822) * q^79 + (-6978*b7 + 13329*b6 - 418*b5 + 1359*b4 - 19873*b3 + 70564*b2 - 4472*b1 + 24337505) * q^81 + (-3815*b7 - 16904*b6 - 1930*b5 + 965*b4 + 17283*b3 + 33579*b2) * q^83 + (-16680*b7 - 132*b6 + 528*b5 + 132*b4 + 132*b3 - 450096*b2 - 18880*b1 + 33041728) * q^85 + (7446*b7 - 24588*b6 + 5646*b5 + 1719*b4 - 60243*b3 + 38046*b2 + 2049*b1 + 30356376) * q^87 + (-6406*b7 - 28165*b6 - 4682*b5 + 2341*b4 + 1609*b3 + 70544*b2) * q^89 + (-9436*b7 + 364*b6 - 1456*b5 - 364*b4 - 364*b3 - 255500*b2 + 30166*b1 + 47766060) * q^91 + (-28662*b7 - 31335*b6 + 8052*b5 - 57*b4 + 19410*b3 + 41057*b2 + 9360*b1 - 39183794) * q^93 + (14142*b7 + 32852*b6 + 5040*b5 - 2520*b4 + 94024*b3 - 362274*b2) * q^95 + (-8754*b7 + 2121*b6 - 8484*b5 - 2121*b4 - 2121*b3 - 240600*b2 - 13852*b1 - 42165838) * q^97 + (-17025*b7 + 31716*b6 - 3940*b5 + 5058*b4 - 49954*b3 - 34877*b2 - 3470*b1 - 46212688) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 56 q^{3} - 1584 q^{7} + 328 q^{9}+O(q^{10})$$ 8 * q + 56 * q^3 - 1584 * q^7 + 328 * q^9 $$8 q + 56 q^{3} - 1584 q^{7} + 328 q^{9} - 25232 q^{13} - 38336 q^{15} + 157936 q^{19} - 30480 q^{21} - 579704 q^{25} - 276040 q^{27} - 805552 q^{31} + 102848 q^{33} + 3985008 q^{37} + 2297104 q^{39} + 6962672 q^{43} - 8670592 q^{45} - 5884520 q^{49} - 15590144 q^{51} - 27101312 q^{55} + 36756688 q^{57} + 51583600 q^{61} + 69759312 q^{63} + 58200688 q^{67} - 94226048 q^{69} - 116854768 q^{73} - 143181896 q^{75} - 172454576 q^{79} + 194700040 q^{81} + 264333824 q^{85} + 242851008 q^{87} + 382128480 q^{91} - 313470352 q^{93} - 337326704 q^{97} - 369701504 q^{99}+O(q^{100})$$ 8 * q + 56 * q^3 - 1584 * q^7 + 328 * q^9 - 25232 * q^13 - 38336 * q^15 + 157936 * q^19 - 30480 * q^21 - 579704 * q^25 - 276040 * q^27 - 805552 * q^31 + 102848 * q^33 + 3985008 * q^37 + 2297104 * q^39 + 6962672 * q^43 - 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 - 27101312 * q^55 + 36756688 * q^57 + 51583600 * q^61 + 69759312 * q^63 + 58200688 * q^67 - 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 - 172454576 * q^79 + 194700040 * q^81 + 264333824 * q^85 + 242851008 * q^87 + 382128480 * q^91 - 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633$$ :

 $$\beta_{1}$$ $$=$$ $$( - 14134 \nu^{7} - 759906 \nu^{6} + 8175192 \nu^{5} + 32119854 \nu^{4} - 356737260 \nu^{3} + \cdots - 10631678772 ) / 6908733$$ (-14134*v^7 - 759906*v^6 + 8175192*v^5 + 32119854*v^4 - 356737260*v^3 - 138768366*v^2 + 1513490626*v - 10631678772) / 6908733 $$\beta_{2}$$ $$=$$ $$( - 90977 \nu^{7} + 306225 \nu^{6} + 8157405 \nu^{5} - 15590670 \nu^{4} - 246555357 \nu^{3} + \cdots + 77404449 ) / 41452398$$ (-90977*v^7 + 306225*v^6 + 8157405*v^5 - 15590670*v^4 - 246555357*v^3 + 22273518*v^2 + 1519133552*v + 77404449) / 41452398 $$\beta_{3}$$ $$=$$ $$( 1348973 \nu^{7} - 7217253 \nu^{6} - 97646277 \nu^{5} + 350373210 \nu^{4} + 2225360013 \nu^{3} + \cdots + 7288059399 ) / 41452398$$ (1348973*v^7 - 7217253*v^6 - 97646277*v^5 + 350373210*v^4 + 2225360013*v^3 - 2487477162*v^2 + 8556056980*v + 7288059399) / 41452398 $$\beta_{4}$$ $$=$$ $$( - 2071745 \nu^{7} + 76404537 \nu^{6} - 163202991 \nu^{5} - 5405297226 \nu^{4} + \cdots - 270823955163 ) / 41452398$$ (-2071745*v^7 + 76404537*v^6 - 163202991*v^5 - 5405297226*v^4 + 7942497303*v^3 + 144605952858*v^2 + 61408458452*v - 270823955163) / 41452398 $$\beta_{5}$$ $$=$$ $$( 1067294 \nu^{7} - 14014314 \nu^{6} - 49856268 \nu^{5} + 883003110 \nu^{4} + 1518931968 \nu^{3} + \cdots - 41475425052 ) / 20726199$$ (1067294*v^7 - 14014314*v^6 - 49856268*v^5 + 883003110*v^4 + 1518931968*v^3 - 16878043182*v^2 - 14759852990*v - 41475425052) / 20726199 $$\beta_{6}$$ $$=$$ $$( 580555 \nu^{7} - 2572731 \nu^{6} - 41619615 \nu^{5} + 91789914 \nu^{4} + 1034741919 \nu^{3} + \cdots + 2039309301 ) / 6908733$$ (580555*v^7 - 2572731*v^6 - 41619615*v^5 + 91789914*v^4 + 1034741919*v^3 + 302019078*v^2 + 2644331312*v + 2039309301) / 6908733 $$\beta_{7}$$ $$=$$ $$( - 17611 \nu^{7} + 68427 \nu^{6} + 1422267 \nu^{5} - 2154414 \nu^{4} - 41574963 \nu^{3} + \cdots - 10880913 ) / 170586$$ (-17611*v^7 + 68427*v^6 + 1422267*v^5 - 2154414*v^4 - 41574963*v^3 - 25987890*v^2 + 113453212*v - 10880913) / 170586
 $$\nu$$ $$=$$ $$( 16\beta_{7} + 23\beta_{6} + 16\beta_{5} + 4\beta_{4} + 4\beta_{3} + 494\beta_{2} - 24\beta _1 + 13824 ) / 27648$$ (16*b7 + 23*b6 + 16*b5 + 4*b4 + 4*b3 + 494*b2 - 24*b1 + 13824) / 27648 $$\nu^{2}$$ $$=$$ $$( 14\beta_{7} + 5\beta_{6} + 10\beta_{5} + 4\beta_{4} - 2\beta_{3} - 370\beta_{2} + 18\beta _1 + 74304 ) / 3456$$ (14*b7 + 5*b6 + 10*b5 + 4*b4 - 2*b3 - 370*b2 + 18*b1 + 74304) / 3456 $$\nu^{3}$$ $$=$$ $$( 296 \beta_{7} + 583 \beta_{6} + 296 \beta_{5} + 56 \beta_{4} - 1024 \beta_{3} - 914 \beta_{2} + \cdots + 654336 ) / 9216$$ (296*b7 + 583*b6 + 296*b5 + 56*b4 - 1024*b3 - 914*b2 - 216*b1 + 654336) / 9216 $$\nu^{4}$$ $$=$$ $$( 3056 \beta_{7} + 2423 \beta_{6} + 1192 \beta_{5} + 358 \beta_{4} - 2186 \beta_{3} - 64282 \beta_{2} + \cdots + 5871744 ) / 6912$$ (3056*b7 + 2423*b6 + 1192*b5 + 358*b4 - 2186*b3 - 64282*b2 + 756*b1 + 5871744) / 6912 $$\nu^{5}$$ $$=$$ $$( 21170 \beta_{7} + 36652 \beta_{6} + 9650 \beta_{5} + 1715 \beta_{4} - 60295 \beta_{3} + \cdots + 29137536 ) / 6912$$ (21170*b7 + 36652*b6 + 9650*b5 + 1715*b4 - 60295*b3 - 297296*b2 - 1992*b1 + 29137536) / 6912 $$\nu^{6}$$ $$=$$ $$( 150456 \beta_{7} + 168071 \beta_{6} + 32536 \beta_{5} + 9064 \beta_{4} - 202784 \beta_{3} + \cdots + 130065408 ) / 4608$$ (150456*b7 + 168071*b6 + 32536*b5 + 9064*b4 - 202784*b3 - 3094642*b2 + 2808*b1 + 130065408) / 4608 $$\nu^{7}$$ $$=$$ $$( 6424576 \beta_{7} + 10532849 \beta_{6} + 1181248 \beta_{5} + 172084 \beta_{4} - 15833948 \beta_{3} + \cdots + 4132200960 ) / 27648$$ (6424576*b7 + 10532849*b6 + 1181248*b5 + 172084*b4 - 15833948*b3 - 122704606*b2 + 214680*b1 + 4132200960) / 27648

## 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
 1.26427 + 1.41421i 1.26427 − 1.41421i 7.73966 − 1.41421i 7.73966 + 1.41421i −5.49843 + 1.41421i −5.49843 − 1.41421i −1.50551 + 1.41421i −1.50551 − 1.41421i
0 −76.4245 26.8384i 0 295.589i 0 843.136 0 5120.40 + 4102.23i 0
65.2 0 −76.4245 + 26.8384i 0 295.589i 0 843.136 0 5120.40 4102.23i 0
65.3 0 −4.95410 80.8484i 0 404.296i 0 262.245 0 −6511.91 + 801.062i 0
65.4 0 −4.95410 + 80.8484i 0 404.296i 0 262.245 0 −6511.91 801.062i 0
65.5 0 28.6420 75.7670i 0 868.404i 0 −3909.88 0 −4920.27 4340.23i 0
65.6 0 28.6420 + 75.7670i 0 868.404i 0 −3909.88 0 −4920.27 + 4340.23i 0
65.7 0 80.7366 6.52720i 0 920.542i 0 2012.50 0 6475.79 1053.97i 0
65.8 0 80.7366 + 6.52720i 0 920.542i 0 2012.50 0 6475.79 + 1053.97i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.8 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.j 8
3.b odd 2 1 inner 192.9.e.j 8
4.b odd 2 1 192.9.e.i 8
8.b even 2 1 48.9.e.e 8
8.d odd 2 1 24.9.e.a 8
12.b even 2 1 192.9.e.i 8
24.f even 2 1 24.9.e.a 8
24.h odd 2 1 48.9.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.e.a 8 8.d odd 2 1
24.9.e.a 8 24.f even 2 1
48.9.e.e 8 8.b even 2 1
48.9.e.e 8 24.h odd 2 1
192.9.e.i 8 4.b odd 2 1
192.9.e.i 8 12.b even 2 1
192.9.e.j 8 1.a even 1 1 trivial
192.9.e.j 8 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 $$S_{9}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{8} + 1852352T_{5}^{6} + 1055033650176T_{5}^{4} + 183162750181376000T_{5}^{2} + 9126569679992258560000$$ T5^8 + 1852352*T5^6 + 1055033650176*T5^4 + 183162750181376000*T5^2 + 9126569679992258560000 $$T_{7}^{4} + 792T_{7}^{3} - 9744840T_{7}^{2} + 9117353952T_{7} - 1739819024496$$ T7^4 + 792*T7^3 - 9744840*T7^2 + 9117353952*T7 - 1739819024496

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + \cdots + 18\!\cdots\!41$$
$5$ $$T^{8} + \cdots + 91\!\cdots\!00$$
$7$ $$(T^{4} + \cdots - 1739819024496)^{2}$$
$11$ $$T^{8} + \cdots + 54\!\cdots\!24$$
$13$ $$(T^{4} + \cdots + 50\!\cdots\!00)^{2}$$
$17$ $$T^{8} + \cdots + 68\!\cdots\!36$$
$19$ $$(T^{4} + \cdots + 14\!\cdots\!56)^{2}$$
$23$ $$T^{8} + \cdots + 10\!\cdots\!56$$
$29$ $$T^{8} + \cdots + 20\!\cdots\!76$$
$31$ $$(T^{4} + \cdots - 24\!\cdots\!88)^{2}$$
$37$ $$(T^{4} + \cdots + 13\!\cdots\!76)^{2}$$
$41$ $$T^{8} + \cdots + 71\!\cdots\!76$$
$43$ $$(T^{4} + \cdots + 40\!\cdots\!44)^{2}$$
$47$ $$T^{8} + \cdots + 14\!\cdots\!56$$
$53$ $$T^{8} + \cdots + 12\!\cdots\!44$$
$59$ $$T^{8} + \cdots + 36\!\cdots\!44$$
$61$ $$(T^{4} + \cdots - 77\!\cdots\!96)^{2}$$
$67$ $$(T^{4} + \cdots + 23\!\cdots\!36)^{2}$$
$71$ $$T^{8} + \cdots + 11\!\cdots\!96$$
$73$ $$(T^{4} + \cdots + 58\!\cdots\!04)^{2}$$
$79$ $$(T^{4} + \cdots - 61\!\cdots\!00)^{2}$$
$83$ $$T^{8} + \cdots + 98\!\cdots\!96$$
$89$ $$T^{8} + \cdots + 25\!\cdots\!36$$
$97$ $$(T^{4} + \cdots - 90\!\cdots\!04)^{2}$$