# Properties

 Label 22.9.d.a Level $22$ Weight $9$ Character orbit 22.d Analytic conductor $8.962$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 22.d (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.96232942134$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10})$$ 32 * q - 182 * q^3 + 1024 * q^4 + 1410 * q^5 + 4480 * q^6 - 10950 * q^7 - 1402 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9} - 47598 q^{11} - 21504 q^{12} + 24990 q^{13} + 106752 q^{14} + 273582 q^{15} - 131072 q^{16} - 553530 q^{17} + 152320 q^{18} + 442680 q^{19} - 180480 q^{20} - 359040 q^{22} - 317064 q^{23} + 573440 q^{24} - 918738 q^{25} - 782592 q^{26} + 714484 q^{27} + 51200 q^{28} + 2380710 q^{29} + 4488960 q^{30} + 181654 q^{31} - 628628 q^{33} - 3865088 q^{34} - 12788370 q^{35} + 5097216 q^{36} - 6976698 q^{37} - 15360 q^{38} + 17382010 q^{39} + 3768320 q^{40} + 5153790 q^{41} - 15355648 q^{42} + 2744064 q^{44} + 5954792 q^{45} - 15447040 q^{46} - 13476378 q^{47} - 2981888 q^{48} + 19896222 q^{49} + 13816320 q^{50} + 37405440 q^{51} + 18368000 q^{52} - 847422 q^{53} - 47212238 q^{55} + 1572864 q^{56} - 119710960 q^{57} - 29125376 q^{58} - 49425120 q^{59} + 12710144 q^{60} + 26432690 q^{61} + 64485120 q^{62} + 289643740 q^{63} + 16777216 q^{64} - 78840064 q^{66} - 133260476 q^{67} - 70851840 q^{68} - 33873776 q^{69} - 50942720 q^{70} + 38304066 q^{71} + 61603840 q^{72} + 208728710 q^{73} + 56355840 q^{74} - 62835924 q^{75} + 92206590 q^{77} - 74988544 q^{78} - 247776110 q^{79} - 22609920 q^{80} - 622273104 q^{81} + 17468928 q^{82} + 434637000 q^{83} + 159336960 q^{84} + 286821150 q^{85} - 7040640 q^{86} + 101498880 q^{88} - 336205452 q^{89} - 270179840 q^{90} - 167007498 q^{91} + 29901312 q^{92} + 65388262 q^{93} + 429944320 q^{94} + 1032904950 q^{95} - 75153432 q^{97} - 459062342 q^{99}+O(q^{100})$$ 32 * q - 182 * q^3 + 1024 * q^4 + 1410 * q^5 + 4480 * q^6 - 10950 * q^7 - 1402 * q^9 - 47598 * q^11 - 21504 * q^12 + 24990 * q^13 + 106752 * q^14 + 273582 * q^15 - 131072 * q^16 - 553530 * q^17 + 152320 * q^18 + 442680 * q^19 - 180480 * q^20 - 359040 * q^22 - 317064 * q^23 + 573440 * q^24 - 918738 * q^25 - 782592 * q^26 + 714484 * q^27 + 51200 * q^28 + 2380710 * q^29 + 4488960 * q^30 + 181654 * q^31 - 628628 * q^33 - 3865088 * q^34 - 12788370 * q^35 + 5097216 * q^36 - 6976698 * q^37 - 15360 * q^38 + 17382010 * q^39 + 3768320 * q^40 + 5153790 * q^41 - 15355648 * q^42 + 2744064 * q^44 + 5954792 * q^45 - 15447040 * q^46 - 13476378 * q^47 - 2981888 * q^48 + 19896222 * q^49 + 13816320 * q^50 + 37405440 * q^51 + 18368000 * q^52 - 847422 * q^53 - 47212238 * q^55 + 1572864 * q^56 - 119710960 * q^57 - 29125376 * q^58 - 49425120 * q^59 + 12710144 * q^60 + 26432690 * q^61 + 64485120 * q^62 + 289643740 * q^63 + 16777216 * q^64 - 78840064 * q^66 - 133260476 * q^67 - 70851840 * q^68 - 33873776 * q^69 - 50942720 * q^70 + 38304066 * q^71 + 61603840 * q^72 + 208728710 * q^73 + 56355840 * q^74 - 62835924 * q^75 + 92206590 * q^77 - 74988544 * q^78 - 247776110 * q^79 - 22609920 * q^80 - 622273104 * q^81 + 17468928 * q^82 + 434637000 * q^83 + 159336960 * q^84 + 286821150 * q^85 - 7040640 * q^86 + 101498880 * q^88 - 336205452 * q^89 - 270179840 * q^90 - 167007498 * q^91 + 29901312 * q^92 + 65388262 * q^93 + 429944320 * q^94 + 1032904950 * q^95 - 75153432 * q^97 - 459062342 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −6.65003 + 9.15298i −24.3639 + 74.9844i −39.5542 121.735i −467.457 + 339.628i −524.310 721.651i −1196.71 + 388.833i 1377.28 + 447.504i 278.905 + 202.637i 6537.16i
7.2 −6.65003 + 9.15298i −20.1856 + 62.1248i −39.5542 121.735i 927.050 673.541i −434.393 597.890i 320.984 104.294i 1377.28 + 447.504i 1855.92 + 1348.41i 12964.3i
7.3 −6.65003 + 9.15298i 15.7928 48.6053i −39.5542 121.735i −394.920 + 286.926i 339.861 + 467.778i 1393.15 452.662i 1377.28 + 447.504i 3194.90 + 2321.23i 5522.77i
7.4 −6.65003 + 9.15298i 39.9571 122.975i −39.5542 121.735i 594.183 431.699i 859.874 + 1183.52i −4362.77 + 1417.55i 1377.28 + 447.504i −8218.38 5971.00i 8309.36i
7.5 6.65003 9.15298i −43.1911 + 132.929i −39.5542 121.735i 371.274 269.746i 929.471 + 1279.31i −2980.04 + 968.273i −1377.28 447.504i −10496.6 7626.22i 5192.08i
7.6 6.65003 9.15298i −17.2507 + 53.0922i −39.5542 121.735i −95.9356 + 69.7013i 371.234 + 510.960i 3856.66 1253.10i −1377.28 447.504i 2786.76 + 2024.70i 1341.61i
7.7 6.65003 9.15298i 10.6908 32.9028i −39.5542 121.735i −864.151 + 627.842i −230.065 316.657i −1443.95 + 469.167i −1377.28 447.504i 4339.66 + 3152.95i 12084.7i
7.8 6.65003 9.15298i 32.1819 99.0456i −39.5542 121.735i 446.808 324.625i −692.552 953.217i 361.475 117.450i −1377.28 447.504i −3466.40 2518.49i 6248.39i
13.1 −10.7600 3.49613i −118.443 + 86.0541i 103.554 + 75.2365i 263.736 + 811.695i 1575.30 511.847i −2234.95 + 3076.14i −851.204 1171.58i 4596.05 14145.2i 9655.88i
13.2 −10.7600 3.49613i −42.1296 + 30.6089i 103.554 + 75.2365i −302.083 929.717i 560.326 182.061i −997.523 + 1372.97i −851.204 1171.58i −1189.47 + 3660.80i 11059.9i
13.3 −10.7600 3.49613i −9.77853 + 7.10452i 103.554 + 75.2365i 24.4502 + 75.2499i 130.055 42.2574i 1395.71 1921.02i −851.204 1171.58i −1982.32 + 6100.94i 895.168i
13.4 −10.7600 3.49613i 67.1008 48.7516i 103.554 + 75.2365i 202.501 + 623.233i −892.445 + 289.973i −159.403 + 219.399i −851.204 1171.58i 98.3408 302.662i 7413.94i
13.5 10.7600 + 3.49613i −124.474 + 90.4358i 103.554 + 75.2365i −172.270 530.193i −1655.51 + 537.909i −92.0512 + 126.698i 851.204 + 1171.58i 5287.73 16274.0i 6307.14i
13.6 10.7600 + 3.49613i −23.2078 + 16.8615i 103.554 + 75.2365i 88.9820 + 273.858i −308.665 + 100.291i −550.185 + 757.265i 851.204 + 1171.58i −1773.17 + 5457.25i 3257.80i
13.7 10.7600 + 3.49613i 43.1191 31.3278i 103.554 + 75.2365i −227.453 700.028i 573.486 186.337i 1739.62 2394.38i 851.204 + 1171.58i −1149.64 + 3538.23i 8327.49i
13.8 10.7600 + 3.49613i 123.182 89.4972i 103.554 + 75.2365i 310.287 + 954.964i 1638.33 532.327i −525.022 + 722.631i 851.204 + 1171.58i 5136.68 15809.1i 11360.2i
17.1 −10.7600 + 3.49613i −118.443 86.0541i 103.554 75.2365i 263.736 811.695i 1575.30 + 511.847i −2234.95 3076.14i −851.204 + 1171.58i 4596.05 + 14145.2i 9655.88i
17.2 −10.7600 + 3.49613i −42.1296 30.6089i 103.554 75.2365i −302.083 + 929.717i 560.326 + 182.061i −997.523 1372.97i −851.204 + 1171.58i −1189.47 3660.80i 11059.9i
17.3 −10.7600 + 3.49613i −9.77853 7.10452i 103.554 75.2365i 24.4502 75.2499i 130.055 + 42.2574i 1395.71 + 1921.02i −851.204 + 1171.58i −1982.32 6100.94i 895.168i
17.4 −10.7600 + 3.49613i 67.1008 + 48.7516i 103.554 75.2365i 202.501 623.233i −892.445 289.973i −159.403 219.399i −851.204 + 1171.58i 98.3408 + 302.662i 7413.94i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.9.d.a 32
11.c even 5 1 242.9.b.e 32
11.d odd 10 1 inner 22.9.d.a 32
11.d odd 10 1 242.9.b.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.9.d.a 32 1.a even 1 1 trivial
22.9.d.a 32 11.d odd 10 1 inner
242.9.b.e 32 11.c even 5 1
242.9.b.e 32 11.d odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(22, [\chi])$$.