Properties

 Label 192.9.g.f Level $192$ Weight $9$ Character orbit 192.g Analytic conductor $78.217$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(127,192)]

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([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (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: $$x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104$$ x^8 - 342*x^6 + 43937*x^4 - 2512824*x^2 + 53993104 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{40}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 96) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{6} + 154) q^{5} + ( - \beta_{4} - \beta_{3} - 9 \beta_{2} - 17 \beta_1) q^{7} - 2187 q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b6 + 154) * q^5 + (-b4 - b3 - 9*b2 - 17*b1) * q^7 - 2187 * q^9 $$q + \beta_1 q^{3} + ( - \beta_{6} + 154) q^{5} + ( - \beta_{4} - \beta_{3} - 9 \beta_{2} - 17 \beta_1) q^{7} - 2187 q^{9} + (22 \beta_{4} - 4 \beta_{3} + 47 \beta_{2} + 56 \beta_1) q^{11} + (\beta_{7} + 13 \beta_{6} + \beta_{5} + 6782) q^{13} + (27 \beta_{4} - 54 \beta_{2} + 166 \beta_1) q^{15} + (5 \beta_{7} + 17 \beta_{6} - 22 \beta_{5} - 25542) q^{17} + ( - 18 \beta_{4} + 38 \beta_{3} + 729 \beta_{2} + 42 \beta_1) q^{19} + ( - 5 \beta_{7} - 76 \beta_{6} - 27 \beta_{5} + 37260) q^{21} + (94 \beta_{4} + 102 \beta_{3} - 1606 \beta_{2} + 1518 \beta_1) q^{23} + ( - 12 \beta_{7} - 296 \beta_{6} + 22 \beta_{5} - 97677) q^{25} - 2187 \beta_1 q^{27} + ( - 120 \beta_{7} + 385 \beta_{6} - 182 \beta_{5} + 41714) q^{29} + ( - 345 \beta_{4} - 325 \beta_{3} - 7679 \beta_{2} + 4675 \beta_1) q^{31} + (115 \beta_{7} + 1667 \beta_{6} - 108 \beta_{5} - 96876) q^{33} + ( - 438 \beta_{4} - 264 \beta_{3} - 17495 \beta_{2} - 13336 \beta_1) q^{35} + ( - 317 \beta_{7} - 383 \beta_{6} + 431 \beta_{5} - 493146) q^{37} + ( - 378 \beta_{4} - 81 \beta_{3} - 1917 \beta_{2} + 6653 \beta_1) q^{39} + (235 \beta_{7} - 5489 \beta_{6} - 246 \beta_{5} + 1475386) q^{41} + (466 \beta_{4} - 394 \beta_{3} - 35073 \beta_{2} - 46014 \beta_1) q^{43} + (2187 \beta_{6} - 336798) q^{45} + ( - 1086 \beta_{4} + 958 \beta_{3} - 1994 \beta_{2} + 57598 \beta_1) q^{47} + (1578 \beta_{7} + 5758 \beta_{6} + 1206 \beta_{5} - 497151) q^{49} + ( - 594 \beta_{4} + 1782 \beta_{3} + 945 \beta_{2} - 26664 \beta_1) q^{51} + ( - 3148 \beta_{7} - 2755 \beta_{6} + 642 \beta_{5} + \cdots - 98574) q^{53}+ \cdots + ( - 48114 \beta_{4} + 8748 \beta_{3} - 102789 \beta_{2} - 122472 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b6 + 154) * q^5 + (-b4 - b3 - 9*b2 - 17*b1) * q^7 - 2187 * q^9 + (22*b4 - 4*b3 + 47*b2 + 56*b1) * q^11 + (b7 + 13*b6 + b5 + 6782) * q^13 + (27*b4 - 54*b2 + 166*b1) * q^15 + (5*b7 + 17*b6 - 22*b5 - 25542) * q^17 + (-18*b4 + 38*b3 + 729*b2 + 42*b1) * q^19 + (-5*b7 - 76*b6 - 27*b5 + 37260) * q^21 + (94*b4 + 102*b3 - 1606*b2 + 1518*b1) * q^23 + (-12*b7 - 296*b6 + 22*b5 - 97677) * q^25 - 2187*b1 * q^27 + (-120*b7 + 385*b6 - 182*b5 + 41714) * q^29 + (-345*b4 - 325*b3 - 7679*b2 + 4675*b1) * q^31 + (115*b7 + 1667*b6 - 108*b5 - 96876) * q^33 + (-438*b4 - 264*b3 - 17495*b2 - 13336*b1) * q^35 + (-317*b7 - 383*b6 + 431*b5 - 493146) * q^37 + (-378*b4 - 81*b3 - 1917*b2 + 6653*b1) * q^39 + (235*b7 - 5489*b6 - 246*b5 + 1475386) * q^41 + (466*b4 - 394*b3 - 35073*b2 - 46014*b1) * q^43 + (2187*b6 - 336798) * q^45 + (-1086*b4 + 958*b3 - 1994*b2 + 57598*b1) * q^47 + (1578*b7 + 5758*b6 + 1206*b5 - 497151) * q^49 + (-594*b4 + 1782*b3 + 945*b2 - 26664*b1) * q^51 + (-3148*b7 - 2755*b6 + 642*b5 - 98574) * q^53 + (5288*b4 - 612*b3 - 48294*b2 + 212068*b1) * q^55 + (465*b7 - 1923*b6 + 1026*b5 - 149364) * q^57 + (4120*b4 + 2356*b3 + 47516*b2 - 202624*b1) * q^59 + (-4829*b7 + 12125*b6 - 1501*b5 + 5031542) * q^61 + (2187*b4 + 2187*b3 + 19683*b2 + 37179*b1) * q^63 + (801*b7 - 5955*b6 + 374*b5 - 2593492) * q^65 + (-3624*b4 - 7996*b3 + 90068*b2 - 264792*b1) * q^67 + (-2030*b7 + 9644*b6 + 2754*b5 - 3335904) * q^69 + (4790*b4 - 2790*b3 + 28870*b2 + 350386*b1) * q^71 + (870*b7 - 4862*b6 - 8198*b5 - 7089006) * q^73 + (8316*b4 - 1782*b3 - 1080*b2 - 93123*b1) * q^75 + (-12374*b7 - 24102*b6 - 3232*b5 + 6501776) * q^77 + (-34741*b4 - 11409*b3 - 37625*b2 + 146311*b1) * q^79 + 4782969 * q^81 + (-23018*b4 + 11440*b3 + 83403*b2 - 448988*b1) * q^83 + (-12096*b7 + 52642*b6 - 3824*b5 - 6587196) * q^85 + (-7155*b4 + 14742*b3 + 365202*b2 + 31436*b1) * q^87 + (-9454*b7 + 64610*b6 + 276*b5 + 7707522) * q^89 + (7930*b4 - 2278*b3 + 269307*b2 + 244410*b1) * q^91 + (-6419*b7 - 21526*b6 - 8775*b5 - 10217340) * q^93 + (-7616*b4 + 18128*b3 + 723424*b2 - 184184*b1) * q^95 + (-32190*b7 - 34302*b6 + 15008*b5 + 3478786) * q^97 + (-48114*b4 + 8748*b3 - 102789*b2 - 122472*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 1232 q^{5} - 17496 q^{9}+O(q^{10})$$ 8 * q + 1232 * q^5 - 17496 * q^9 $$8 q + 1232 q^{5} - 17496 q^{9} + 54256 q^{13} - 204336 q^{17} + 298080 q^{21} - 781416 q^{25} + 333712 q^{29} - 775008 q^{33} - 3945168 q^{37} + 11803088 q^{41} - 2694384 q^{45} - 3977208 q^{49} - 788592 q^{53} - 1194912 q^{57} + 40252336 q^{61} - 20747936 q^{65} - 26687232 q^{69} - 56712048 q^{73} + 52014208 q^{77} + 38263752 q^{81} - 52697568 q^{85} + 61660176 q^{89} - 81738720 q^{93} + 27830288 q^{97}+O(q^{100})$$ 8 * q + 1232 * q^5 - 17496 * q^9 + 54256 * q^13 - 204336 * q^17 + 298080 * q^21 - 781416 * q^25 + 333712 * q^29 - 775008 * q^33 - 3945168 * q^37 + 11803088 * q^41 - 2694384 * q^45 - 3977208 * q^49 - 788592 * q^53 - 1194912 * q^57 + 40252336 * q^61 - 20747936 * q^65 - 26687232 * q^69 - 56712048 * q^73 + 52014208 * q^77 + 38263752 * q^81 - 52697568 * q^85 + 61660176 * q^89 - 81738720 * q^93 + 27830288 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104$$ :

 $$\beta_{1}$$ $$=$$ $$( -27\nu^{7} + 9234\nu^{5} - 987903\nu^{3} + 33920532\nu ) / 58784$$ (-27*v^7 + 9234*v^5 - 987903*v^3 + 33920532*v) / 58784 $$\beta_{2}$$ $$=$$ $$( 348\nu^{7} - 89624\nu^{5} + 7765724\nu^{3} - 226222192\nu ) / 123079$$ (348*v^7 - 89624*v^5 + 7765724*v^3 - 226222192*v) / 123079 $$\beta_{3}$$ $$=$$ $$( - 871 \nu^{7} + 235136 \nu^{6} + 297882 \nu^{5} - 123211264 \nu^{4} - 31869019 \nu^{3} + 15898720640 \nu^{2} + 1094251236 \nu - 607942246912 ) / 3938528$$ (-871*v^7 + 235136*v^6 + 297882*v^5 - 123211264*v^4 - 31869019*v^3 + 15898720640*v^2 + 1094251236*v - 607942246912) / 3938528 $$\beta_{4}$$ $$=$$ $$( 27 \nu^{7} - 352704 \nu^{6} - 23930 \nu^{5} + 90292224 \nu^{4} + 3471527 \nu^{3} - 7778886720 \nu^{2} - 139408420 \nu + 225427939968 ) / 984632$$ (27*v^7 - 352704*v^6 - 23930*v^5 + 90292224*v^4 + 3471527*v^3 - 7778886720*v^2 - 139408420*v + 225427939968) / 984632 $$\beta_{5}$$ $$=$$ $$( - 1044 \nu^{7} + 44088 \nu^{6} + 268872 \nu^{5} - 11286528 \nu^{4} - 23297172 \nu^{3} + 960545256 \nu^{2} + 725928912 \nu - 27168260064 ) / 123079$$ (-1044*v^7 + 44088*v^6 + 268872*v^5 - 11286528*v^4 - 23297172*v^3 + 960545256*v^2 + 725928912*v - 27168260064) / 123079 $$\beta_{6}$$ $$=$$ $$( - 14562 \nu^{7} - 20207 \nu^{6} + 3716348 \nu^{5} + 5172992 \nu^{4} - 315278826 \nu^{3} - 440249909 \nu^{2} + 8887740344 \nu + 12452119196 ) / 492316$$ (-14562*v^7 - 20207*v^6 + 3716348*v^5 + 5172992*v^4 - 315278826*v^3 - 440249909*v^2 + 8887740344*v + 12452119196) / 492316 $$\beta_{7}$$ $$=$$ $$( - 14562 \nu^{7} + 773377 \nu^{6} + 3716348 \nu^{5} - 197984512 \nu^{4} - 315278826 \nu^{3} + 16849564699 \nu^{2} + 8887740344 \nu - 476576561956 ) / 492316$$ (-14562*v^7 + 773377*v^6 + 3716348*v^5 - 197984512*v^4 - 315278826*v^3 + 16849564699*v^2 + 8887740344*v - 476576561956) / 492316
 $$\nu$$ $$=$$ $$( -2\beta_{7} + 2\beta_{6} + 9\beta_{5} + 27\beta_{2} ) / 3456$$ (-2*b7 + 2*b6 + 9*b5 + 27*b2) / 3456 $$\nu^{2}$$ $$=$$ $$( -32\beta_{7} + 32\beta_{6} - 144\beta_{4} - 9\beta_{2} - 64\beta _1 + 1181952 ) / 13824$$ (-32*b7 + 32*b6 - 144*b4 - 9*b2 - 64*b1 + 1181952) / 13824 $$\nu^{3}$$ $$=$$ $$( -53\beta_{7} + 197\beta_{6} + 255\beta_{5} + 2313\beta_{2} + 256\beta_1 ) / 1152$$ (-53*b7 + 197*b6 + 255*b5 + 2313*b2 + 256*b1) / 1152 $$\nu^{4}$$ $$=$$ $$( -5440\beta_{7} + 5440\beta_{6} - 24624\beta_{4} - 864\beta_{3} - 1539\beta_{2} - 10528\beta _1 + 100535040 ) / 13824$$ (-5440*b7 + 5440*b6 - 24624*b4 - 864*b3 - 1539*b2 - 10528*b1 + 100535040) / 13824 $$\nu^{5}$$ $$=$$ $$( -12515\beta_{7} + 85523\beta_{6} + 64683\beta_{5} + 993357\beta_{2} + 218880\beta_1 ) / 3456$$ (-12515*b7 + 85523*b6 + 64683*b5 + 993357*b2 + 218880*b1) / 3456 $$\nu^{6}$$ $$=$$ $$( - 228960 \beta_{7} + 228960 \beta_{6} - 1055472 \beta_{4} - 73728 \beta_{3} - 65967 \beta_{2} - 433600 \beta _1 + 2834839296 ) / 4608$$ (-228960*b7 + 228960*b6 - 1055472*b4 - 73728*b3 - 65967*b2 - 433600*b1 + 2834839296) / 4608 $$\nu^{7}$$ $$=$$ $$( -975111\beta_{7} + 10137399\beta_{6} + 5437845\beta_{5} + 119757555\beta_{2} + 39232256\beta_1 ) / 3456$$ (-975111*b7 + 10137399*b6 + 5437845*b5 + 119757555*b2 + 39232256*b1) / 3456

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 9.62695 − 0.500000i −8.87816 + 0.500000i −9.62695 − 0.500000i 8.87816 + 0.500000i 9.62695 + 0.500000i −8.87816 − 0.500000i −9.62695 + 0.500000i 8.87816 − 0.500000i
0 46.7654i 0 −455.790 0 1689.76i 0 −2187.00 0
127.2 0 46.7654i 0 −261.868 0 1340.75i 0 −2187.00 0
127.3 0 46.7654i 0 611.370 0 4427.25i 0 −2187.00 0
127.4 0 46.7654i 0 722.288 0 891.264i 0 −2187.00 0
127.5 0 46.7654i 0 −455.790 0 1689.76i 0 −2187.00 0
127.6 0 46.7654i 0 −261.868 0 1340.75i 0 −2187.00 0
127.7 0 46.7654i 0 611.370 0 4427.25i 0 −2187.00 0
127.8 0 46.7654i 0 722.288 0 891.264i 0 −2187.00 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.g.f 8
4.b odd 2 1 inner 192.9.g.f 8
8.b even 2 1 96.9.g.a 8
8.d odd 2 1 96.9.g.a 8
24.f even 2 1 288.9.g.f 8
24.h odd 2 1 288.9.g.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.g.a 8 8.b even 2 1
96.9.g.a 8 8.d odd 2 1
192.9.g.f 8 1.a even 1 1 trivial
192.9.g.f 8 4.b odd 2 1 inner
288.9.g.f 8 24.f even 2 1
288.9.g.f 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 616T_{5}^{3} - 396168T_{5}^{2} + 157725920T_{5} + 52706093200$$ acting on $$S_{9}^{\mathrm{new}}(192, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 2187)^{4}$$
$5$ $$(T^{4} - 616 T^{3} + \cdots + 52706093200)^{2}$$
$7$ $$T^{8} + 25047808 T^{6} + \cdots + 79\!\cdots\!84$$
$11$ $$T^{8} + 1747174336 T^{6} + \cdots + 65\!\cdots\!84$$
$13$ $$(T^{4} - 27128 T^{3} + \cdots - 81\!\cdots\!44)^{2}$$
$17$ $$(T^{4} + 102168 T^{3} + \cdots + 28\!\cdots\!68)^{2}$$
$19$ $$T^{8} + 33336733632 T^{6} + \cdots + 10\!\cdots\!84$$
$23$ $$T^{8} + 277565694208 T^{6} + \cdots + 21\!\cdots\!76$$
$29$ $$(T^{4} - 166856 T^{3} + \cdots - 40\!\cdots\!92)^{2}$$
$31$ $$T^{8} + 3440896309504 T^{6} + \cdots + 13\!\cdots\!76$$
$37$ $$(T^{4} + 1972584 T^{3} + \cdots + 30\!\cdots\!84)^{2}$$
$41$ $$(T^{4} - 5901544 T^{3} + \cdots - 37\!\cdots\!12)^{2}$$
$43$ $$T^{8} + 42117576693184 T^{6} + \cdots + 70\!\cdots\!00$$
$47$ $$T^{8} + 48015145188352 T^{6} + \cdots + 26\!\cdots\!16$$
$53$ $$(T^{4} + 394296 T^{3} + \cdots + 34\!\cdots\!72)^{2}$$
$59$ $$T^{8} + 556916566724800 T^{6} + \cdots + 31\!\cdots\!64$$
$61$ $$(T^{4} - 20126168 T^{3} + \cdots + 19\!\cdots\!00)^{2}$$
$67$ $$T^{8} + \cdots + 41\!\cdots\!96$$
$71$ $$T^{8} + \cdots + 13\!\cdots\!16$$
$73$ $$(T^{4} + 28356024 T^{3} + \cdots + 60\!\cdots\!52)^{2}$$
$79$ $$T^{8} + \cdots + 16\!\cdots\!24$$
$83$ $$T^{8} + \cdots + 54\!\cdots\!16$$
$89$ $$(T^{4} - 30830088 T^{3} + \cdots - 54\!\cdots\!04)^{2}$$
$97$ $$(T^{4} - 13915144 T^{3} + \cdots - 66\!\cdots\!12)^{2}$$