# Properties

 Label 192.11.e.h Level $192$ Weight $11$ Character orbit 192.e Analytic conductor $121.989$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$121.988592513$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{85})$$ Defining polynomial: $$x^{4} - 2x^{3} - 37x^{2} + 38x + 531$$ x^4 - 2*x^3 - 37*x^2 + 38*x + 531 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{14}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 21) q^{3} + (\beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} - 43 \beta_1 + 11278) q^{7} + (7 \beta_{3} - 62 \beta_{2} - 20 \beta_1 + 39753) q^{9}+O(q^{10})$$ q + (b1 + 21) * q^3 + (b3 + 2*b2 - 4*b1) * q^5 + (-b3 + 5*b2 - 43*b1 + 11278) * q^7 + (7*b3 - 62*b2 - 20*b1 + 39753) * q^9 $$q + (\beta_1 + 21) q^{3} + (\beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} - 43 \beta_1 + 11278) q^{7} + (7 \beta_{3} - 62 \beta_{2} - 20 \beta_1 + 39753) q^{9} + (35 \beta_{3} + 189 \beta_{2} - 21 \beta_1) q^{11} + ( - 20 \beta_{3} + 100 \beta_{2} - 860 \beta_1 - 68810) q^{13} + (35 \beta_{3} + 1877 \beta_{2} + 743 \beta_1 + 295200) q^{15} + ( - 236 \beta_{3} + 2860 \beta_{2} + 4276 \beta_1) q^{17} + ( - 97 \beta_{3} + 485 \beta_{2} - 4171 \beta_1 - 392182) q^{19} + ( - 189 \beta_{3} + 1674 \beta_{2} + 12952 \beta_1 - 2407002) q^{21} + ( - 826 \beta_{3} - 4590 \beta_{2} + 366 \beta_1) q^{23} + ( - 420 \beta_{3} + 2100 \beta_{2} - 18060 \beta_1 - 8433095) q^{25} + ( - 1599 \beta_{3} + 5727 \beta_{2} + 39306 \beta_1 + 8654877) q^{27} + (2947 \beta_{3} + 34494 \beta_{2} + 16812 \beta_1) q^{29} + ( - 3045 \beta_{3} + 15225 \beta_{2} - 130935 \beta_1 + 5446462) q^{31} + (5033 \beta_{3} + 69146 \beta_{2} + 29456 \beta_1 + 6493536) q^{33} + (9598 \beta_{3} - 71914 \beta_{2} - 129502 \beta_1) q^{35} + (5684 \beta_{3} - 28420 \beta_{2} + 244412 \beta_1 + 17753542) q^{37} + ( - 3780 \beta_{3} + 33480 \beta_{2} - 35330 \beta_1 - 54321810) q^{39} + ( - 13414 \beta_{3} + 158308 \beta_{2} + 238792 \beta_1) q^{41} + (6783 \beta_{3} - 33915 \beta_{2} + 291669 \beta_1 - 117672166) q^{43} + (52581 \beta_{3} + 175386 \beta_{2} + 411492 \beta_1 - 78079680) q^{45} + (14460 \beta_{3} - 159940 \beta_{2} - 246700 \beta_1) q^{47} + ( - 22556 \beta_{3} + 112780 \beta_{2} - 969908 \beta_1 - 12514605) q^{49} + (98364 \beta_{3} - 346344 \beta_{2} - 78720 \beta_1 - 177144192) q^{51} + ( - 59339 \beta_{3} - 437750 \beta_{2} - 81716 \beta_1) q^{53} + ( - 37548 \beta_{3} + 187740 \beta_{2} + \cdots - 675339840) q^{55}+ \cdots + (2000271 \beta_{3} + 11748765 \beta_{2} + \cdots - 656728128) q^{99}+O(q^{100})$$ q + (b1 + 21) * q^3 + (b3 + 2*b2 - 4*b1) * q^5 + (-b3 + 5*b2 - 43*b1 + 11278) * q^7 + (7*b3 - 62*b2 - 20*b1 + 39753) * q^9 + (35*b3 + 189*b2 - 21*b1) * q^11 + (-20*b3 + 100*b2 - 860*b1 - 68810) * q^13 + (35*b3 + 1877*b2 + 743*b1 + 295200) * q^15 + (-236*b3 + 2860*b2 + 4276*b1) * q^17 + (-97*b3 + 485*b2 - 4171*b1 - 392182) * q^19 + (-189*b3 + 1674*b2 + 12952*b1 - 2407002) * q^21 + (-826*b3 - 4590*b2 + 366*b1) * q^23 + (-420*b3 + 2100*b2 - 18060*b1 - 8433095) * q^25 + (-1599*b3 + 5727*b2 + 39306*b1 + 8654877) * q^27 + (2947*b3 + 34494*b2 + 16812*b1) * q^29 + (-3045*b3 + 15225*b2 - 130935*b1 + 5446462) * q^31 + (5033*b3 + 69146*b2 + 29456*b1 + 6493536) * q^33 + (9598*b3 - 71914*b2 - 129502*b1) * q^35 + (5684*b3 - 28420*b2 + 244412*b1 + 17753542) * q^37 + (-3780*b3 + 33480*b2 - 35330*b1 - 54321810) * q^39 + (-13414*b3 + 158308*b2 + 238792*b1) * q^41 + (6783*b3 - 33915*b2 + 291669*b1 - 117672166) * q^43 + (52581*b3 + 175386*b2 + 411492*b1 - 78079680) * q^45 + (14460*b3 - 159940*b2 - 246700*b1) * q^47 + (-22556*b3 + 112780*b2 - 969908*b1 - 12514605) * q^49 + (98364*b3 - 346344*b2 - 78720*b1 - 177144192) * q^51 + (-59339*b3 - 437750*b2 - 81716*b1) * q^53 + (-37548*b3 + 187740*b2 - 1614564*b1 - 675339840) * q^55 + (-18333*b3 + 162378*b2 - 229804*b1 - 264688302) * q^57 + (-23273*b3 - 739875*b2 - 600237*b1) * q^59 + (-19644*b3 + 98220*b2 - 844692*b1 + 296009686) * q^61 + (130057*b3 - 924173*b2 - 2903825*b1 + 226251774) * q^63 + (-102410*b3 - 2027020*b2 - 1412560*b1) * q^65 + (-49917*b3 + 249585*b2 - 2146431*b1 - 74341462) * q^67 + (-122926*b3 - 1635604*b2 - 698920*b1 - 149067072) * q^69 + (-213598*b3 - 803026*b2 + 478562*b1) * q^71 + (-46224*b3 + 231120*b2 - 1987632*b1 + 1633567250) * q^73 + (-79380*b3 + 703080*b2 - 7730015*b1 - 1287507795) * q^75 + (153146*b3 - 1340556*b2 - 2259432*b1) * q^77 + (160363*b3 - 801815*b2 + 6895609*b1 - 49820642) * q^79 + (397530*b3 - 4229568*b2 + 6742260*b1 + 364741137) * q^81 + (-388359*b3 + 651583*b2 + 2981737*b1) * q^83 + (-540624*b3 + 2703120*b2 - 23246832*b1 + 3220128000) * q^85 + (1018345*b3 + 6360919*b2 + 3019021*b1 - 52567200) * q^87 + (587062*b3 + 5538400*b2 + 2016028*b1) * q^89 + (-156750*b3 + 783750*b2 - 6740250*b1 + 2079308020) * q^91 + (-575505*b3 + 5097330*b2 + 10543792*b1 - 7936117098) * q^93 + (-555142*b3 - 9947954*b2 - 6617102*b1) * q^95 + (-79756*b3 + 398780*b2 - 3429508*b1 - 9794088766) * q^97 + (2000271*b3 + 11748765*b2 + 12335547*b1 - 656728128) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 84 q^{3} + 45112 q^{7} + 159012 q^{9}+O(q^{10})$$ 4 * q + 84 * q^3 + 45112 * q^7 + 159012 * q^9 $$4 q + 84 q^{3} + 45112 q^{7} + 159012 q^{9} - 275240 q^{13} + 1180800 q^{15} - 1568728 q^{19} - 9628008 q^{21} - 33732380 q^{25} + 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} + 71014168 q^{37} - 217287240 q^{39} - 470688664 q^{43} - 312318720 q^{45} - 50058420 q^{49} - 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} + 1184038744 q^{61} + 905007096 q^{63} - 297365848 q^{67} - 596268288 q^{69} + 6534269000 q^{73} - 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} + 12880512000 q^{85} - 210268800 q^{87} + 8317232080 q^{91} - 31744468392 q^{93} - 39176355064 q^{97} - 2626912512 q^{99}+O(q^{100})$$ 4 * q + 84 * q^3 + 45112 * q^7 + 159012 * q^9 - 275240 * q^13 + 1180800 * q^15 - 1568728 * q^19 - 9628008 * q^21 - 33732380 * q^25 + 34619508 * q^27 + 21785848 * q^31 + 25974144 * q^33 + 71014168 * q^37 - 217287240 * q^39 - 470688664 * q^43 - 312318720 * q^45 - 50058420 * q^49 - 708576768 * q^51 - 2701359360 * q^55 - 1058753208 * q^57 + 1184038744 * q^61 + 905007096 * q^63 - 297365848 * q^67 - 596268288 * q^69 + 6534269000 * q^73 - 5150031180 * q^75 - 199282568 * q^79 + 1458964548 * q^81 + 12880512000 * q^85 - 210268800 * q^87 + 8317232080 * q^91 - 31744468392 * q^93 - 39176355064 * q^97 - 2626912512 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 37x^{2} + 38x + 531$$ :

 $$\beta_{1}$$ $$=$$ $$( -60\nu^{3} + 276\nu^{2} + 2172\nu - 4728 ) / 31$$ (-60*v^3 + 276*v^2 + 2172*v - 4728) / 31 $$\beta_{2}$$ $$=$$ $$( -196\nu^{3} + 108\nu^{2} + 1540\nu + 2808 ) / 31$$ (-196*v^3 + 108*v^2 + 1540*v + 2808) / 31 $$\beta_{3}$$ $$=$$ $$( -128\nu^{3} - 8736\nu^{2} + 19712\nu + 164208 ) / 31$$ (-128*v^3 - 8736*v^2 + 19712*v + 164208) / 31
 $$\nu$$ $$=$$ $$( 4\beta_{3} - 47\beta_{2} + 145\beta _1 + 5184 ) / 10368$$ (4*b3 - 47*b2 + 145*b1 + 5184) / 10368 $$\nu^{2}$$ $$=$$ $$( -7\beta_{3} - 19\beta_{2} + 77\beta _1 + 50544 ) / 2592$$ (-7*b3 - 19*b2 + 77*b1 + 50544) / 2592 $$\nu^{3}$$ $$=$$ $$( 16\beta_{3} - 2051\beta_{2} + 1309\beta _1 + 300672 ) / 10368$$ (16*b3 - 2051*b2 + 1309*b1 + 300672) / 10368

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −4.10977 + 1.41421i −4.10977 − 1.41421i 5.10977 − 1.41421i 5.10977 + 1.41421i
0 −200.269 137.627i 0 3630.47i 0 23226.5 0 21166.4 + 55125.0i 0
65.2 0 −200.269 + 137.627i 0 3630.47i 0 23226.5 0 21166.4 55125.0i 0
65.3 0 242.269 18.8335i 0 4818.41i 0 −670.530 0 58339.6 9125.53i 0
65.4 0 242.269 + 18.8335i 0 4818.41i 0 −670.530 0 58339.6 + 9125.53i 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 192.11.e.h 4
3.b odd 2 1 inner 192.11.e.h 4
4.b odd 2 1 192.11.e.g 4
8.b even 2 1 48.11.e.d 4
8.d odd 2 1 6.11.b.a 4
12.b even 2 1 192.11.e.g 4
24.f even 2 1 6.11.b.a 4
24.h odd 2 1 48.11.e.d 4
40.e odd 2 1 150.11.d.a 4
40.k even 4 2 150.11.b.a 8
72.l even 6 2 162.11.d.d 8
72.p odd 6 2 162.11.d.d 8
120.m even 2 1 150.11.d.a 4
120.q odd 4 2 150.11.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 8.d odd 2 1
6.11.b.a 4 24.f even 2 1
48.11.e.d 4 8.b even 2 1
48.11.e.d 4 24.h odd 2 1
150.11.b.a 8 40.k even 4 2
150.11.b.a 8 120.q odd 4 2
150.11.d.a 4 40.e odd 2 1
150.11.d.a 4 120.m even 2 1
162.11.d.d 8 72.l even 6 2
162.11.d.d 8 72.p odd 6 2
192.11.e.g 4 4.b odd 2 1
192.11.e.g 4 12.b even 2 1
192.11.e.h 4 1.a even 1 1 trivial
192.11.e.h 4 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 $$S_{11}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{4} + 36397440T_{5}^{2} + 306009247334400$$ T5^4 + 36397440*T5^2 + 306009247334400 $$T_{7}^{2} - 22556T_{7} - 15574076$$ T7^2 - 22556*T7 - 15574076

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 84 T^{3} + \cdots + 3486784401$$
$5$ $$T^{4} + \cdots + 306009247334400$$
$7$ $$(T^{2} - 22556 T - 15574076)^{2}$$
$11$ $$T^{4} + 58313211264 T^{2} + \cdots + 21\!\cdots\!24$$
$13$ $$(T^{2} + 137620 T - 52372127900)^{2}$$
$17$ $$T^{4} + 7560967182336 T^{2} + \cdots + 32\!\cdots\!84$$
$19$ $$(T^{2} + 784364 T - 1189491369116)^{2}$$
$23$ $$T^{4} + 33055507478016 T^{2} + \cdots + 61\!\cdots\!24$$
$29$ $$T^{4} + 907304099736960 T^{2} + \cdots + 20\!\cdots\!00$$
$31$ $$(T^{2} - 10892924 T - 12\!\cdots\!56)^{2}$$
$37$ $$(T^{2} - 35507084 T - 42\!\cdots\!96)^{2}$$
$41$ $$T^{4} + \cdots + 28\!\cdots\!04$$
$43$ $$(T^{2} + 235344332 T + 72\!\cdots\!16)^{2}$$
$47$ $$T^{4} + \cdots + 26\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 37\!\cdots\!04$$
$59$ $$T^{4} + \cdots + 20\!\cdots\!04$$
$61$ $$(T^{2} - 592019372 T + 32\!\cdots\!36)^{2}$$
$67$ $$(T^{2} + 148682924 T - 35\!\cdots\!96)^{2}$$
$71$ $$T^{4} + \cdots + 49\!\cdots\!00$$
$73$ $$(T^{2} - 3267134500 T + 23\!\cdots\!40)^{2}$$
$79$ $$(T^{2} + 99641284 T - 36\!\cdots\!76)^{2}$$
$83$ $$T^{4} + \cdots + 57\!\cdots\!44$$
$89$ $$T^{4} + \cdots + 15\!\cdots\!84$$
$97$ $$(T^{2} + 19588177532 T + 95\!\cdots\!96)^{2}$$