# Properties

 Label 22.8.a.d Level $22$ Weight $8$ Character orbit 22.a Self dual yes Analytic conductor $6.872$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 22.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.87247056065$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14881})$$ Defining polynomial: $$x^{2} - x - 3720$$ x^2 - x - 3720 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{14881})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 q^{2} + ( - \beta - 11) q^{3} + 64 q^{4} + (\beta + 165) q^{5} + ( - 8 \beta - 88) q^{6} + (14 \beta + 890) q^{7} + 512 q^{8} + (23 \beta + 1654) q^{9}+O(q^{10})$$ q + 8 * q^2 + (-b - 11) * q^3 + 64 * q^4 + (b + 165) * q^5 + (-8*b - 88) * q^6 + (14*b + 890) * q^7 + 512 * q^8 + (23*b + 1654) * q^9 $$q + 8 q^{2} + ( - \beta - 11) q^{3} + 64 q^{4} + (\beta + 165) q^{5} + ( - 8 \beta - 88) q^{6} + (14 \beta + 890) q^{7} + 512 q^{8} + (23 \beta + 1654) q^{9} + (8 \beta + 1320) q^{10} + 1331 q^{11} + ( - 64 \beta - 704) q^{12} + ( - 118 \beta - 2644) q^{13} + (112 \beta + 7120) q^{14} + ( - 177 \beta - 5535) q^{15} + 4096 q^{16} + (236 \beta + 7398) q^{17} + (184 \beta + 13232) q^{18} + (484 \beta + 8216) q^{19} + (64 \beta + 10560) q^{20} + ( - 1058 \beta - 61870) q^{21} + 10648 q^{22} + (567 \beta - 25959) q^{23} + ( - 512 \beta - 5632) q^{24} + (331 \beta - 47180) q^{25} + ( - 944 \beta - 21152) q^{26} + (257 \beta - 79697) q^{27} + (896 \beta + 56960) q^{28} + ( - 730 \beta - 103200) q^{29} + ( - 1416 \beta - 44280) q^{30} + (1295 \beta - 10183) q^{31} + 32768 q^{32} + ( - 1331 \beta - 14641) q^{33} + (1888 \beta + 59184) q^{34} + (3214 \beta + 198930) q^{35} + (1472 \beta + 105856) q^{36} + ( - 3385 \beta + 177359) q^{37} + (3872 \beta + 65728) q^{38} + (4060 \beta + 468044) q^{39} + (512 \beta + 84480) q^{40} + ( - 8006 \beta + 65808) q^{41} + ( - 8464 \beta - 494960) q^{42} + ( - 12682 \beta - 73570) q^{43} + 85184 q^{44} + (5472 \beta + 358470) q^{45} + (4536 \beta - 207672) q^{46} + (5768 \beta + 222696) q^{47} + ( - 4096 \beta - 45056) q^{48} + (25116 \beta + 697677) q^{49} + (2648 \beta - 377440) q^{50} + ( - 10230 \beta - 959298) q^{51} + ( - 7552 \beta - 169216) q^{52} + (9308 \beta - 635070) q^{53} + (2056 \beta - 637576) q^{54} + (1331 \beta + 219615) q^{55} + (7168 \beta + 455680) q^{56} + ( - 14024 \beta - 1890856) q^{57} + ( - 5840 \beta - 825600) q^{58} + ( - 14307 \beta + 450927) q^{59} + ( - 11328 \beta - 354240) q^{60} + ( - 12118 \beta - 292900) q^{61} + (10360 \beta - 81464) q^{62} + (43948 \beta + 2669900) q^{63} + 262144 q^{64} + ( - 22232 \beta - 875220) q^{65} + ( - 10648 \beta - 117128) q^{66} + ( - 34943 \beta + 1449827) q^{67} + (15104 \beta + 473472) q^{68} + (19155 \beta - 1823691) q^{69} + (25712 \beta + 1591440) q^{70} + ( - 40763 \beta + 673515) q^{71} + (11776 \beta + 846848) q^{72} + (29326 \beta - 2303428) q^{73} + ( - 27080 \beta + 1418872) q^{74} + (43208 \beta - 712340) q^{75} + (30976 \beta + 525824) q^{76} + (18634 \beta + 1184590) q^{77} + (32480 \beta + 3744352) q^{78} + ( - 55258 \beta - 1445542) q^{79} + (4096 \beta + 675840) q^{80} + (26312 \beta - 3696671) q^{81} + ( - 64048 \beta + 526464) q^{82} + ( - 24754 \beta + 4995102) q^{83} + ( - 67712 \beta - 3959680) q^{84} + (46574 \beta + 2098590) q^{85} + ( - 101456 \beta - 588560) q^{86} + (111960 \beta + 3850800) q^{87} + 681472 q^{88} + ( - 67837 \beta + 5126607) q^{89} + (43776 \beta + 2867760) q^{90} + ( - 143688 \beta - 8498600) q^{91} + (36288 \beta - 1661376) q^{92} + ( - 5357 \beta - 4705387) q^{93} + (46144 \beta + 1781568) q^{94} + (88560 \beta + 3156120) q^{95} + ( - 32768 \beta - 360448) q^{96} + ( - 1255 \beta - 13882111) q^{97} + (200928 \beta + 5581416) q^{98} + (30613 \beta + 2201474) q^{99}+O(q^{100})$$ q + 8 * q^2 + (-b - 11) * q^3 + 64 * q^4 + (b + 165) * q^5 + (-8*b - 88) * q^6 + (14*b + 890) * q^7 + 512 * q^8 + (23*b + 1654) * q^9 + (8*b + 1320) * q^10 + 1331 * q^11 + (-64*b - 704) * q^12 + (-118*b - 2644) * q^13 + (112*b + 7120) * q^14 + (-177*b - 5535) * q^15 + 4096 * q^16 + (236*b + 7398) * q^17 + (184*b + 13232) * q^18 + (484*b + 8216) * q^19 + (64*b + 10560) * q^20 + (-1058*b - 61870) * q^21 + 10648 * q^22 + (567*b - 25959) * q^23 + (-512*b - 5632) * q^24 + (331*b - 47180) * q^25 + (-944*b - 21152) * q^26 + (257*b - 79697) * q^27 + (896*b + 56960) * q^28 + (-730*b - 103200) * q^29 + (-1416*b - 44280) * q^30 + (1295*b - 10183) * q^31 + 32768 * q^32 + (-1331*b - 14641) * q^33 + (1888*b + 59184) * q^34 + (3214*b + 198930) * q^35 + (1472*b + 105856) * q^36 + (-3385*b + 177359) * q^37 + (3872*b + 65728) * q^38 + (4060*b + 468044) * q^39 + (512*b + 84480) * q^40 + (-8006*b + 65808) * q^41 + (-8464*b - 494960) * q^42 + (-12682*b - 73570) * q^43 + 85184 * q^44 + (5472*b + 358470) * q^45 + (4536*b - 207672) * q^46 + (5768*b + 222696) * q^47 + (-4096*b - 45056) * q^48 + (25116*b + 697677) * q^49 + (2648*b - 377440) * q^50 + (-10230*b - 959298) * q^51 + (-7552*b - 169216) * q^52 + (9308*b - 635070) * q^53 + (2056*b - 637576) * q^54 + (1331*b + 219615) * q^55 + (7168*b + 455680) * q^56 + (-14024*b - 1890856) * q^57 + (-5840*b - 825600) * q^58 + (-14307*b + 450927) * q^59 + (-11328*b - 354240) * q^60 + (-12118*b - 292900) * q^61 + (10360*b - 81464) * q^62 + (43948*b + 2669900) * q^63 + 262144 * q^64 + (-22232*b - 875220) * q^65 + (-10648*b - 117128) * q^66 + (-34943*b + 1449827) * q^67 + (15104*b + 473472) * q^68 + (19155*b - 1823691) * q^69 + (25712*b + 1591440) * q^70 + (-40763*b + 673515) * q^71 + (11776*b + 846848) * q^72 + (29326*b - 2303428) * q^73 + (-27080*b + 1418872) * q^74 + (43208*b - 712340) * q^75 + (30976*b + 525824) * q^76 + (18634*b + 1184590) * q^77 + (32480*b + 3744352) * q^78 + (-55258*b - 1445542) * q^79 + (4096*b + 675840) * q^80 + (26312*b - 3696671) * q^81 + (-64048*b + 526464) * q^82 + (-24754*b + 4995102) * q^83 + (-67712*b - 3959680) * q^84 + (46574*b + 2098590) * q^85 + (-101456*b - 588560) * q^86 + (111960*b + 3850800) * q^87 + 681472 * q^88 + (-67837*b + 5126607) * q^89 + (43776*b + 2867760) * q^90 + (-143688*b - 8498600) * q^91 + (36288*b - 1661376) * q^92 + (-5357*b - 4705387) * q^93 + (46144*b + 1781568) * q^94 + (88560*b + 3156120) * q^95 + (-32768*b - 360448) * q^96 + (-1255*b - 13882111) * q^97 + (200928*b + 5581416) * q^98 + (30613*b + 2201474) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9}+O(q^{10})$$ 2 * q + 16 * q^2 - 23 * q^3 + 128 * q^4 + 331 * q^5 - 184 * q^6 + 1794 * q^7 + 1024 * q^8 + 3331 * q^9 $$2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9} + 2648 q^{10} + 2662 q^{11} - 1472 q^{12} - 5406 q^{13} + 14352 q^{14} - 11247 q^{15} + 8192 q^{16} + 15032 q^{17} + 26648 q^{18} + 16916 q^{19} + 21184 q^{20} - 124798 q^{21} + 21296 q^{22} - 51351 q^{23} - 11776 q^{24} - 94029 q^{25} - 43248 q^{26} - 159137 q^{27} + 114816 q^{28} - 207130 q^{29} - 89976 q^{30} - 19071 q^{31} + 65536 q^{32} - 30613 q^{33} + 120256 q^{34} + 401074 q^{35} + 213184 q^{36} + 351333 q^{37} + 135328 q^{38} + 940148 q^{39} + 169472 q^{40} + 123610 q^{41} - 998384 q^{42} - 159822 q^{43} + 170368 q^{44} + 722412 q^{45} - 410808 q^{46} + 451160 q^{47} - 94208 q^{48} + 1420470 q^{49} - 752232 q^{50} - 1928826 q^{51} - 345984 q^{52} - 1260832 q^{53} - 1273096 q^{54} + 440561 q^{55} + 918528 q^{56} - 3795736 q^{57} - 1657040 q^{58} + 887547 q^{59} - 719808 q^{60} - 597918 q^{61} - 152568 q^{62} + 5383748 q^{63} + 524288 q^{64} - 1772672 q^{65} - 244904 q^{66} + 2864711 q^{67} + 962048 q^{68} - 3628227 q^{69} + 3208592 q^{70} + 1306267 q^{71} + 1705472 q^{72} - 4577530 q^{73} + 2810664 q^{74} - 1381472 q^{75} + 1082624 q^{76} + 2387814 q^{77} + 7521184 q^{78} - 2946342 q^{79} + 1355776 q^{80} - 7367030 q^{81} + 988880 q^{82} + 9965450 q^{83} - 7987072 q^{84} + 4243754 q^{85} - 1278576 q^{86} + 7813560 q^{87} + 1362944 q^{88} + 10185377 q^{89} + 5779296 q^{90} - 17140888 q^{91} - 3286464 q^{92} - 9416131 q^{93} + 3609280 q^{94} + 6400800 q^{95} - 753664 q^{96} - 27765477 q^{97} + 11363760 q^{98} + 4433561 q^{99}+O(q^{100})$$ 2 * q + 16 * q^2 - 23 * q^3 + 128 * q^4 + 331 * q^5 - 184 * q^6 + 1794 * q^7 + 1024 * q^8 + 3331 * q^9 + 2648 * q^10 + 2662 * q^11 - 1472 * q^12 - 5406 * q^13 + 14352 * q^14 - 11247 * q^15 + 8192 * q^16 + 15032 * q^17 + 26648 * q^18 + 16916 * q^19 + 21184 * q^20 - 124798 * q^21 + 21296 * q^22 - 51351 * q^23 - 11776 * q^24 - 94029 * q^25 - 43248 * q^26 - 159137 * q^27 + 114816 * q^28 - 207130 * q^29 - 89976 * q^30 - 19071 * q^31 + 65536 * q^32 - 30613 * q^33 + 120256 * q^34 + 401074 * q^35 + 213184 * q^36 + 351333 * q^37 + 135328 * q^38 + 940148 * q^39 + 169472 * q^40 + 123610 * q^41 - 998384 * q^42 - 159822 * q^43 + 170368 * q^44 + 722412 * q^45 - 410808 * q^46 + 451160 * q^47 - 94208 * q^48 + 1420470 * q^49 - 752232 * q^50 - 1928826 * q^51 - 345984 * q^52 - 1260832 * q^53 - 1273096 * q^54 + 440561 * q^55 + 918528 * q^56 - 3795736 * q^57 - 1657040 * q^58 + 887547 * q^59 - 719808 * q^60 - 597918 * q^61 - 152568 * q^62 + 5383748 * q^63 + 524288 * q^64 - 1772672 * q^65 - 244904 * q^66 + 2864711 * q^67 + 962048 * q^68 - 3628227 * q^69 + 3208592 * q^70 + 1306267 * q^71 + 1705472 * q^72 - 4577530 * q^73 + 2810664 * q^74 - 1381472 * q^75 + 1082624 * q^76 + 2387814 * q^77 + 7521184 * q^78 - 2946342 * q^79 + 1355776 * q^80 - 7367030 * q^81 + 988880 * q^82 + 9965450 * q^83 - 7987072 * q^84 + 4243754 * q^85 - 1278576 * q^86 + 7813560 * q^87 + 1362944 * q^88 + 10185377 * q^89 + 5779296 * q^90 - 17140888 * q^91 - 3286464 * q^92 - 9416131 * q^93 + 3609280 * q^94 + 6400800 * q^95 - 753664 * q^96 - 27765477 * q^97 + 11363760 * q^98 + 4433561 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 61.4939 −60.4939
8.00000 −72.4939 64.0000 226.494 −579.951 1750.91 512.000 3068.36 1811.95
1.2 8.00000 49.4939 64.0000 104.506 395.951 43.0861 512.000 262.641 836.049
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.8.a.d 2
3.b odd 2 1 198.8.a.f 2
4.b odd 2 1 176.8.a.e 2
5.b even 2 1 550.8.a.d 2
11.b odd 2 1 242.8.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.8.a.d 2 1.a even 1 1 trivial
176.8.a.e 2 4.b odd 2 1
198.8.a.f 2 3.b odd 2 1
242.8.a.h 2 11.b odd 2 1
550.8.a.d 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 23T_{3} - 3588$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(22))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 8)^{2}$$
$3$ $$T^{2} + 23T - 3588$$
$5$ $$T^{2} - 331T + 23670$$
$7$ $$T^{2} - 1794T + 75440$$
$11$ $$(T - 1331)^{2}$$
$13$ $$T^{2} + 5406 T - 44494552$$
$17$ $$T^{2} - 15032 T - 150712788$$
$19$ $$T^{2} - 16916 T - 799953120$$
$23$ $$T^{2} + 51351 T - 536788152$$
$29$ $$T^{2} + 207130 T + 8743188000$$
$31$ $$T^{2} + 19071 T - 6148026496$$
$37$ $$T^{2} - 351333 T - 11768742334$$
$41$ $$T^{2} - 123610 T - 234633419904$$
$43$ $$T^{2} + 159822 T - 591953661640$$
$47$ $$T^{2} - 451160 T - 72885726336$$
$53$ $$T^{2} + 1260832 T + 75106099260$$
$59$ $$T^{2} - 887547 T - 564563979540$$
$61$ $$T^{2} + 597918 T - 456927065080$$
$67$ $$T^{2} - 2864711 T - 2490832261212$$
$71$ $$T^{2} - 1306267 T - 5755066505400$$
$73$ $$T^{2} + 4577530 T + 2038977114936$$
$79$ $$T^{2} + 2946342 T - 9189351784480$$
$83$ $$T^{2} - 9965450 T + 22547926115976$$
$89$ $$T^{2} - 10185377 T + 8815411816710$$
$97$ $$T^{2} + \cdots + 192724568772626$$