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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9} - 47598 q^{11} - 21504 q^{12} + 24990 q^{13} + 106752 q^{14} + 273582 q^{15} - 131072 q^{16} - 553530 q^{17} + 152320 q^{18} + 442680 q^{19} - 180480 q^{20} - 359040 q^{22} - 317064 q^{23} + 573440 q^{24} - 918738 q^{25} - 782592 q^{26} + 714484 q^{27} + 51200 q^{28} + 2380710 q^{29} + 4488960 q^{30} + 181654 q^{31} - 628628 q^{33} - 3865088 q^{34} - 12788370 q^{35} + 5097216 q^{36} - 6976698 q^{37} - 15360 q^{38} + 17382010 q^{39} + 3768320 q^{40} + 5153790 q^{41} - 15355648 q^{42} + 2744064 q^{44} + 5954792 q^{45} - 15447040 q^{46} - 13476378 q^{47} - 2981888 q^{48} + 19896222 q^{49} + 13816320 q^{50} + 37405440 q^{51} + 18368000 q^{52} - 847422 q^{53} - 47212238 q^{55} + 1572864 q^{56} - 119710960 q^{57} - 29125376 q^{58} - 49425120 q^{59} + 12710144 q^{60} + 26432690 q^{61} + 64485120 q^{62} + 289643740 q^{63} + 16777216 q^{64} - 78840064 q^{66} - 133260476 q^{67} - 70851840 q^{68} - 33873776 q^{69} - 50942720 q^{70} + 38304066 q^{71} + 61603840 q^{72} + 208728710 q^{73} + 56355840 q^{74} - 62835924 q^{75} + 92206590 q^{77} - 74988544 q^{78} - 247776110 q^{79} - 22609920 q^{80} - 622273104 q^{81} + 17468928 q^{82} + 434637000 q^{83} + 159336960 q^{84} + 286821150 q^{85} - 7040640 q^{86} + 101498880 q^{88} - 336205452 q^{89} - 270179840 q^{90} - 167007498 q^{91} + 29901312 q^{92} + 65388262 q^{93} + 429944320 q^{94} + 1032904950 q^{95} - 75153432 q^{97} - 459062342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:

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 \(S_{9}^{\mathrm{new}}(22, [\chi])\).