# Properties

 Label 22.12.a.a Level $22$ Weight $12$ Character orbit 22.a Self dual yes Analytic conductor $16.904$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 22.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.9035499723$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{331})$$ Defining polynomial: $$x^{2} - 331$$ x^2 - 331 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8\sqrt{331}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 32 q^{2} + (3 \beta - 213) q^{3} + 1024 q^{4} + ( - 50 \beta + 1145) q^{5} + (96 \beta - 6816) q^{6} + ( - 165 \beta - 43162) q^{7} + 32768 q^{8} + ( - 1278 \beta + 58878) q^{9}+O(q^{10})$$ q + 32 * q^2 + (3*b - 213) * q^3 + 1024 * q^4 + (-50*b + 1145) * q^5 + (96*b - 6816) * q^6 + (-165*b - 43162) * q^7 + 32768 * q^8 + (-1278*b + 58878) * q^9 $$q + 32 q^{2} + (3 \beta - 213) q^{3} + 1024 q^{4} + ( - 50 \beta + 1145) q^{5} + (96 \beta - 6816) q^{6} + ( - 165 \beta - 43162) q^{7} + 32768 q^{8} + ( - 1278 \beta + 58878) q^{9} + ( - 1600 \beta + 36640) q^{10} - 161051 q^{11} + (3072 \beta - 218112) q^{12} + (8839 \beta - 1050092) q^{13} + ( - 5280 \beta - 1381184) q^{14} + (14085 \beta - 3421485) q^{15} + 1048576 q^{16} + (19951 \beta - 1441138) q^{17} + ( - 40896 \beta + 1884096) q^{18} + (6371 \beta - 9785856) q^{19} + ( - 51200 \beta + 1172480) q^{20} + ( - 94341 \beta - 1292574) q^{21} - 5153632 q^{22} + (106463 \beta + 6340267) q^{23} + (98304 \beta - 6979584) q^{24} + ( - 114500 \beta + 5442900) q^{25} + (282848 \beta - 33602944) q^{26} + ( - 82593 \beta - 56028159) q^{27} + ( - 168960 \beta - 44197888) q^{28} + ( - 420342 \beta + 22831248) q^{29} + (450720 \beta - 109487520) q^{30} + (190135 \beta - 253252085) q^{31} + 33554432 q^{32} + ( - 483153 \beta + 34303863) q^{33} + (638432 \beta - 46116416) q^{34} + (1969175 \beta + 125347510) q^{35} + ( - 1308672 \beta + 60291072) q^{36} + (2302978 \beta + 201336259) q^{37} + (203872 \beta - 313147392) q^{38} + ( - 5032983 \beta + 785405724) q^{39} + ( - 1638400 \beta + 37519360) q^{40} + ( - 6012597 \beta + 304432008) q^{41} + ( - 3018912 \beta - 41362368) q^{42} + (8085116 \beta + 550047282) q^{43} - 164916224 q^{44} + ( - 4407210 \beta + 1421072910) q^{45} + (3406816 \beta + 202888544) q^{46} + (14560338 \beta + 506171136) q^{47} + (3145728 \beta - 223346688) q^{48} + (14243460 \beta + 462365901) q^{49} + ( - 3664000 \beta + 174172800) q^{50} + ( - 8572977 \beta + 1574888346) q^{51} + (9051136 \beta - 1075294208) q^{52} + ( - 24738110 \beta - 34094638) q^{53} + ( - 2642976 \beta - 1792901088) q^{54} + (8052550 \beta - 184403395) q^{55} + ( - 5406720 \beta - 1414332416) q^{56} + ( - 30714591 \beta + 2489277120) q^{57} + ( - 13450944 \beta + 730599936) q^{58} + (22140099 \beta - 3395808759) q^{59} + (14423040 \beta - 3503600640) q^{60} + ( - 57000010 \beta + 2852023260) q^{61} + (6084320 \beta - 8104066720) q^{62} + (45446166 \beta + 1925777844) q^{63} + 1073741824 q^{64} + (62625255 \beta - 10564624140) q^{65} + ( - 15460896 \beta + 1097723616) q^{66} + ( - 9427609 \beta - 18257155851) q^{67} + (20429824 \beta - 1475725312) q^{68} + ( - 3655818 \beta + 5415459705) q^{69} + (63013600 \beta + 4011120320) q^{70} + ( - 115678293 \beta + 10336098297) q^{71} + ( - 41877504 \beta + 1929314304) q^{72} + (51530311 \beta + 1541435428) q^{73} + (73695296 \beta + 6442760288) q^{74} + (40717200 \beta - 8436041700) q^{75} + (6523904 \beta - 10020716544) q^{76} + (26573415 \beta + 6951283262) q^{77} + ( - 161055456 \beta + 25132983168) q^{78} + ( - 2327652 \beta - 14340691090) q^{79} + ( - 52428800 \beta + 1200619520) q^{80} + (75901698 \beta - 3745013535) q^{81} + ( - 192403104 \beta + 9741824256) q^{82} + ( - 68251162 \beta + 14766320386) q^{83} + ( - 96605184 \beta - 1323595776) q^{84} + (94900795 \beta - 22782202210) q^{85} + (258723712 \beta + 17601513024) q^{86} + (158026590 \beta - 31576630608) q^{87} - 5277319168 q^{88} + (35615342 \beta + 42531871231) q^{89} + ( - 141030720 \beta + 45474333120) q^{90} + ( - 208243738 \beta + 14428583864) q^{91} + (109018112 \beta + 6492433408) q^{92} + ( - 800255010 \beta + 66026153625) q^{93} + (465930816 \beta + 16197476352) q^{94} + (496587595 \beta - 17952968320) q^{95} + (100663296 \beta - 7147094016) q^{96} + ( - 149119060 \beta - 90416224839) q^{97} + (455790720 \beta + 14795708832) q^{98} + (205823178 \beta - 9482360778) q^{99}+O(q^{100})$$ q + 32 * q^2 + (3*b - 213) * q^3 + 1024 * q^4 + (-50*b + 1145) * q^5 + (96*b - 6816) * q^6 + (-165*b - 43162) * q^7 + 32768 * q^8 + (-1278*b + 58878) * q^9 + (-1600*b + 36640) * q^10 - 161051 * q^11 + (3072*b - 218112) * q^12 + (8839*b - 1050092) * q^13 + (-5280*b - 1381184) * q^14 + (14085*b - 3421485) * q^15 + 1048576 * q^16 + (19951*b - 1441138) * q^17 + (-40896*b + 1884096) * q^18 + (6371*b - 9785856) * q^19 + (-51200*b + 1172480) * q^20 + (-94341*b - 1292574) * q^21 - 5153632 * q^22 + (106463*b + 6340267) * q^23 + (98304*b - 6979584) * q^24 + (-114500*b + 5442900) * q^25 + (282848*b - 33602944) * q^26 + (-82593*b - 56028159) * q^27 + (-168960*b - 44197888) * q^28 + (-420342*b + 22831248) * q^29 + (450720*b - 109487520) * q^30 + (190135*b - 253252085) * q^31 + 33554432 * q^32 + (-483153*b + 34303863) * q^33 + (638432*b - 46116416) * q^34 + (1969175*b + 125347510) * q^35 + (-1308672*b + 60291072) * q^36 + (2302978*b + 201336259) * q^37 + (203872*b - 313147392) * q^38 + (-5032983*b + 785405724) * q^39 + (-1638400*b + 37519360) * q^40 + (-6012597*b + 304432008) * q^41 + (-3018912*b - 41362368) * q^42 + (8085116*b + 550047282) * q^43 - 164916224 * q^44 + (-4407210*b + 1421072910) * q^45 + (3406816*b + 202888544) * q^46 + (14560338*b + 506171136) * q^47 + (3145728*b - 223346688) * q^48 + (14243460*b + 462365901) * q^49 + (-3664000*b + 174172800) * q^50 + (-8572977*b + 1574888346) * q^51 + (9051136*b - 1075294208) * q^52 + (-24738110*b - 34094638) * q^53 + (-2642976*b - 1792901088) * q^54 + (8052550*b - 184403395) * q^55 + (-5406720*b - 1414332416) * q^56 + (-30714591*b + 2489277120) * q^57 + (-13450944*b + 730599936) * q^58 + (22140099*b - 3395808759) * q^59 + (14423040*b - 3503600640) * q^60 + (-57000010*b + 2852023260) * q^61 + (6084320*b - 8104066720) * q^62 + (45446166*b + 1925777844) * q^63 + 1073741824 * q^64 + (62625255*b - 10564624140) * q^65 + (-15460896*b + 1097723616) * q^66 + (-9427609*b - 18257155851) * q^67 + (20429824*b - 1475725312) * q^68 + (-3655818*b + 5415459705) * q^69 + (63013600*b + 4011120320) * q^70 + (-115678293*b + 10336098297) * q^71 + (-41877504*b + 1929314304) * q^72 + (51530311*b + 1541435428) * q^73 + (73695296*b + 6442760288) * q^74 + (40717200*b - 8436041700) * q^75 + (6523904*b - 10020716544) * q^76 + (26573415*b + 6951283262) * q^77 + (-161055456*b + 25132983168) * q^78 + (-2327652*b - 14340691090) * q^79 + (-52428800*b + 1200619520) * q^80 + (75901698*b - 3745013535) * q^81 + (-192403104*b + 9741824256) * q^82 + (-68251162*b + 14766320386) * q^83 + (-96605184*b - 1323595776) * q^84 + (94900795*b - 22782202210) * q^85 + (258723712*b + 17601513024) * q^86 + (158026590*b - 31576630608) * q^87 - 5277319168 * q^88 + (35615342*b + 42531871231) * q^89 + (-141030720*b + 45474333120) * q^90 + (-208243738*b + 14428583864) * q^91 + (109018112*b + 6492433408) * q^92 + (-800255010*b + 66026153625) * q^93 + (465930816*b + 16197476352) * q^94 + (496587595*b - 17952968320) * q^95 + (100663296*b - 7147094016) * q^96 + (-149119060*b - 90416224839) * q^97 + (455790720*b + 14795708832) * q^98 + (205823178*b - 9482360778) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 64 q^{2} - 426 q^{3} + 2048 q^{4} + 2290 q^{5} - 13632 q^{6} - 86324 q^{7} + 65536 q^{8} + 117756 q^{9}+O(q^{10})$$ 2 * q + 64 * q^2 - 426 * q^3 + 2048 * q^4 + 2290 * q^5 - 13632 * q^6 - 86324 * q^7 + 65536 * q^8 + 117756 * q^9 $$2 q + 64 q^{2} - 426 q^{3} + 2048 q^{4} + 2290 q^{5} - 13632 q^{6} - 86324 q^{7} + 65536 q^{8} + 117756 q^{9} + 73280 q^{10} - 322102 q^{11} - 436224 q^{12} - 2100184 q^{13} - 2762368 q^{14} - 6842970 q^{15} + 2097152 q^{16} - 2882276 q^{17} + 3768192 q^{18} - 19571712 q^{19} + 2344960 q^{20} - 2585148 q^{21} - 10307264 q^{22} + 12680534 q^{23} - 13959168 q^{24} + 10885800 q^{25} - 67205888 q^{26} - 112056318 q^{27} - 88395776 q^{28} + 45662496 q^{29} - 218975040 q^{30} - 506504170 q^{31} + 67108864 q^{32} + 68607726 q^{33} - 92232832 q^{34} + 250695020 q^{35} + 120582144 q^{36} + 402672518 q^{37} - 626294784 q^{38} + 1570811448 q^{39} + 75038720 q^{40} + 608864016 q^{41} - 82724736 q^{42} + 1100094564 q^{43} - 329832448 q^{44} + 2842145820 q^{45} + 405777088 q^{46} + 1012342272 q^{47} - 446693376 q^{48} + 924731802 q^{49} + 348345600 q^{50} + 3149776692 q^{51} - 2150588416 q^{52} - 68189276 q^{53} - 3585802176 q^{54} - 368806790 q^{55} - 2828664832 q^{56} + 4978554240 q^{57} + 1461199872 q^{58} - 6791617518 q^{59} - 7007201280 q^{60} + 5704046520 q^{61} - 16208133440 q^{62} + 3851555688 q^{63} + 2147483648 q^{64} - 21129248280 q^{65} + 2195447232 q^{66} - 36514311702 q^{67} - 2951450624 q^{68} + 10830919410 q^{69} + 8022240640 q^{70} + 20672196594 q^{71} + 3858628608 q^{72} + 3082870856 q^{73} + 12885520576 q^{74} - 16872083400 q^{75} - 20041433088 q^{76} + 13902566524 q^{77} + 50265966336 q^{78} - 28681382180 q^{79} + 2401239040 q^{80} - 7490027070 q^{81} + 19483648512 q^{82} + 29532640772 q^{83} - 2647191552 q^{84} - 45564404420 q^{85} + 35203026048 q^{86} - 63153261216 q^{87} - 10554638336 q^{88} + 85063742462 q^{89} + 90948666240 q^{90} + 28857167728 q^{91} + 12984866816 q^{92} + 132052307250 q^{93} + 32394952704 q^{94} - 35905936640 q^{95} - 14294188032 q^{96} - 180832449678 q^{97} + 29591417664 q^{98} - 18964721556 q^{99}+O(q^{100})$$ 2 * q + 64 * q^2 - 426 * q^3 + 2048 * q^4 + 2290 * q^5 - 13632 * q^6 - 86324 * q^7 + 65536 * q^8 + 117756 * q^9 + 73280 * q^10 - 322102 * q^11 - 436224 * q^12 - 2100184 * q^13 - 2762368 * q^14 - 6842970 * q^15 + 2097152 * q^16 - 2882276 * q^17 + 3768192 * q^18 - 19571712 * q^19 + 2344960 * q^20 - 2585148 * q^21 - 10307264 * q^22 + 12680534 * q^23 - 13959168 * q^24 + 10885800 * q^25 - 67205888 * q^26 - 112056318 * q^27 - 88395776 * q^28 + 45662496 * q^29 - 218975040 * q^30 - 506504170 * q^31 + 67108864 * q^32 + 68607726 * q^33 - 92232832 * q^34 + 250695020 * q^35 + 120582144 * q^36 + 402672518 * q^37 - 626294784 * q^38 + 1570811448 * q^39 + 75038720 * q^40 + 608864016 * q^41 - 82724736 * q^42 + 1100094564 * q^43 - 329832448 * q^44 + 2842145820 * q^45 + 405777088 * q^46 + 1012342272 * q^47 - 446693376 * q^48 + 924731802 * q^49 + 348345600 * q^50 + 3149776692 * q^51 - 2150588416 * q^52 - 68189276 * q^53 - 3585802176 * q^54 - 368806790 * q^55 - 2828664832 * q^56 + 4978554240 * q^57 + 1461199872 * q^58 - 6791617518 * q^59 - 7007201280 * q^60 + 5704046520 * q^61 - 16208133440 * q^62 + 3851555688 * q^63 + 2147483648 * q^64 - 21129248280 * q^65 + 2195447232 * q^66 - 36514311702 * q^67 - 2951450624 * q^68 + 10830919410 * q^69 + 8022240640 * q^70 + 20672196594 * q^71 + 3858628608 * q^72 + 3082870856 * q^73 + 12885520576 * q^74 - 16872083400 * q^75 - 20041433088 * q^76 + 13902566524 * q^77 + 50265966336 * q^78 - 28681382180 * q^79 + 2401239040 * q^80 - 7490027070 * q^81 + 19483648512 * q^82 + 29532640772 * q^83 - 2647191552 * q^84 - 45564404420 * q^85 + 35203026048 * q^86 - 63153261216 * q^87 - 10554638336 * q^88 + 85063742462 * q^89 + 90948666240 * q^90 + 28857167728 * q^91 + 12984866816 * q^92 + 132052307250 * q^93 + 32394952704 * q^94 - 35905936640 * q^95 - 14294188032 * q^96 - 180832449678 * q^97 + 29591417664 * q^98 - 18964721556 * 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
 −18.1934 18.1934
32.0000 −649.642 1024.00 8422.36 −20788.5 −19146.7 32768.0 244887. 269516.
1.2 32.0000 223.642 1024.00 −6132.36 7156.54 −67177.3 32768.0 −127131. −196236.
 $$n$$: 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 22.12.a.a 2
3.b odd 2 1 198.12.a.c 2
4.b odd 2 1 176.12.a.a 2
11.b odd 2 1 242.12.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.a.a 2 1.a even 1 1 trivial
176.12.a.a 2 4.b odd 2 1
198.12.a.c 2 3.b odd 2 1
242.12.a.a 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 426T_{3} - 145287$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(22))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 32)^{2}$$
$3$ $$T^{2} + 426T - 145287$$
$5$ $$T^{2} - 2290 T - 51648975$$
$7$ $$T^{2} + 86324 T + 1286223844$$
$11$ $$(T + 161051)^{2}$$
$13$ $$T^{2} + 2100184 T - 552368670000$$
$17$ $$T^{2} + 2882276 T - 6355251487740$$
$19$ $$T^{2} + 19571712 T + 94903126697792$$
$23$ $$T^{2} + \cdots - 199908316265607$$
$29$ $$T^{2} - 45662496 T - 32\!\cdots\!72$$
$31$ $$T^{2} + 506504170 T + 63\!\cdots\!25$$
$37$ $$T^{2} - 402672518 T - 71\!\cdots\!75$$
$41$ $$T^{2} - 608864016 T - 67\!\cdots\!92$$
$43$ $$T^{2} - 1100094564 T - 10\!\cdots\!80$$
$47$ $$T^{2} - 1012342272 T - 42\!\cdots\!00$$
$53$ $$T^{2} + 68189276 T - 12\!\cdots\!56$$
$59$ $$T^{2} + 6791617518 T + 11\!\cdots\!97$$
$61$ $$T^{2} - 5704046520 T - 60\!\cdots\!00$$
$67$ $$T^{2} + 36514311702 T + 33\!\cdots\!97$$
$71$ $$T^{2} - 20672196594 T - 17\!\cdots\!07$$
$73$ $$T^{2} - 3082870856 T - 53\!\cdots\!80$$
$79$ $$T^{2} + 28681382180 T + 20\!\cdots\!64$$
$83$ $$T^{2} - 29532640772 T + 11\!\cdots\!00$$
$89$ $$T^{2} - 85063742462 T + 17\!\cdots\!85$$
$97$ $$T^{2} + 180832449678 T + 77\!\cdots\!21$$